At its most basic, a fork is when one copy of a project diverges from another copy: think "fork in the road".
What such divergence means for the project depends on the intentions behind the fork. There are two types of forks: development forks and hard forks. The distinction between them is important.
Development forks are very common; in fact, they are the normal way development is done in most projects today. A developer creates her own public copy of the project's master repository, makes some changes, then submits the changes back to the project directly from the forked copy. Development forks are done on a routine basis as part of the expected contribution cycle, and have no negative effect on the social cohesiveness of the project. They are really just an extension of the concept of development branches.
Hard forks (also sometimes called social forks) are much less common, and are much more significant when they happen. A hard fork is when a group of developers is unhappy with the direction of the project and decides to create a divergent version more in line with their own vision. Of course, one of the technical actions required for such this is to create their own copy of the project's repository, and perhaps of its bug database and other assets as well. But this new copy of the project represents a potentially permanent divergence, and developers on both sides of the fork are aware of this; thus, it is a completely different beast from a cooperative development fork.
A hard fork is almost always accompanied by long discussions and rationales, in which developers try to persuade each other of the merits of one or the other side of the fork, or of the merits of ending the fork and reunifying. Since hard forks have implications for a project's stability and ability to continue attracting developers, knowing how to constructively initiate or react to a hard fork of your project is useful — useful even if a fork never happens, as understanding what leads to hard forks, and signalling clearly how you will behave in such an event, can sometimes prevent the fork from happening in the first place.
The rest of this section is about hard forks, not development forks. To save space, I will just use the word "fork" instead of "hard fork".
In the section called “Forkability”, we saw how the potential to fork has important effects on how projects are governed. But what happens when a fork actually occurs? How should you handle it, and what effects can you expect it to have? Conversely, when should you initiate a fork?
The answers depend on what kind of fork it is. Some forks are due to amicable but irreconcilable disagreements about the direction of the project; perhaps more are due to both technical disagreements and interpersonal conflicts. Of course, it's not always possible to tell the difference between the two, as technical arguments may involve personal elements as well. What all forks have in common is that one group of developers (or sometimes even just one developer) has decided that the costs of working with some or all of the others now outweigh the benefits.
Once a project forks, there is no definitive answer to the question of which fork is the "true" or "original" project. People will colloquially talk of fork F coming out of project P, as though P is continuing unchanged down some natural path while F diverges into new territory, but this is, in effect, a declaration of how that particular observer feels about it. It is fundamentally a matter of perception: when a large enough percentage of observers agree, the assertion starts to become true. It is not the case that there is an objective truth from the outset, one that we are only imperfectly able to perceive at first. Rather, the perceptions are the objective truth, since ultimately a project — or a fork — is an entity that exists only in people's minds anyway.
If those initiating the fork feel that they are sprouting a new branch off the main project, the perception question is resolved immediately and easily. Everyone, both developers and users, will treat the fork as a new project, with a new name (perhaps based on the old name, but easily distinguishable from it), a separate web site, and a separate philosophy or goal. Things get messier, however, when both sides feel they are the legitimate guardians of the original project and therefore have the right to continue using the original name. If there is some organization with trademark rights to the name (see the section called “Trademarks”), or legal control over the domain or web pages, that usually resolves the issue by fiat: that organization will decide who is the original project and who is the fork, because it holds all the cards in a public relations war. Naturally, things rarely get that far: since everyone already knows what the power dynamics are, they will avoid fighting a battle whose outcome is known in advance, and will just jump straight to the end instead.
Fortunately, in most cases there is little doubt as to which is the project and which is the fork, because a fork is, in essence, a vote of confidence. If more than half of the developers are in favor of whatever course the fork proposes to take, usually there is no need to fork — the project can simply go that way itself, unless it is run as a dictatorship with a particularly stubborn dictator. On the other hand, if fewer than half of the developers are in favor, the fork is a clearly minority rebellion, and both courtesy and common sense indicate that it should think of itself as the divergent branch rather than the main line.
When a fork occurs, there can be a question of what happens to non-copyable assets: not just trademarks, but perhaps money in the bank, hardware, that full-color conference banner sitting in a storage locker somewhere, etc. Sometimes, those questions are resolved independently of the project's decision-making procedures, because those assets already had formal owners, and in each case the owner will decide what happens to the asset. But in cases where the actual ownership is in dispute, or the asset belongs in some way to the project as a whole, there is no magic answer. If someone decides to make a fuss, the dispute might wind up in a court of law. In this respect, open source projects are not different from any other endeavor involving multiple people: when agreement cannot be reached but no one is willing to give in, the last resort is the legal system. It is extremely rare, however, for things to go that far in a free software project (I can't think of any examples, actually), because usually there is no participant for whom going to court is a better option than just giving up their side of the argument anyway.
If someone threatens a fork in your project, keep calm and remember your long-term goals. The mere existence of a fork isn't what hurts a project; rather, it's the loss of developers and users. Your real aim, therefore, is not to squelch the fork, but to minimize these harmful effects. You may be mad, you may feel that the fork was unjust and uncalled for, but expressing that publicly can only alienate undecided developers. Instead, don't force people to make exclusive choices, and be as cooperative as is practicable with the fork.
Don't remove someone's commit access in your project just because she decided to work on the fork. Work on the fork doesn't mean that person has suddenly lost her competence to work on the original project; committers before should remain committers afterward. Beyond that, you should express your desire to remain as compatible as possible with the fork, and say that you hope developers will port changes between the two whenever appropriate. If you have administrative access to the project's servers, publicly offer the forkers infrastructure help at startup time. For example, offer them a complete export of the bug database if there's no other way for them to get it. Ask them if there's anything else they need, and provide it if you can. Bend over backward to show that you are not standing in the way, and that you want the fork to succeed or fail on its own merits and nothing else.
The reason to do all this — and do it publicly — is not to actually help the fork, but to persuade developers that your side is a safe bet, by appearing as non-vindictive as possible. In war it sometimes makes sense (strategic sense, if not human sense) to force people to choose sides, but in free software it almost never does. In fact, after a fork some developers often openly work on both projects, doing their best to keep the two compatible. These developers help keep the lines of communication open after the fork. They allow your project to benefit from interesting new features in the fork (yes, the fork may have things you want), and also increase the chances of a merger down the road.
Sometimes a fork becomes so successful that, even though it was regarded even by its own instigators as a fork at the outset, it becomes the version everybody prefers, and eventually supplants the original by popular demand. A famous instance of this was the GCC/EGCS fork. The GNU Compiler Collection (GCC, formerly the GNU C Compiler) is the most popular open source native-code compiler, and also one of the most portable compilers in the world. Due to disagreements between the GCC's official maintainers and Cygnus Software, one of GCC's most active developer groups, Cygnus created a fork of GCC called EGCS. The fork was deliberately non-adversarial: the EGCS developers did not, at any point, try to portray their version of GCC as a new official version. Instead, they concentrated on making EGCS as good as possible, incorporating patches at a faster rate than the official GCC maintainers. EGCS grew in popularity, and eventually some major operating system distributors decided to package EGCS as their default compiler instead of GCC. At this point, it became clear to the GCC maintainers that holding on to the "GCC" name while everyone switched to the EGCS fork would burden everyone with a needless name change, yet do nothing to prevent the switchover. So GCC adopted the EGCS codebase, and there is once again a single GCC, but greatly improved because of the fork.
This example shows why you cannot always regard a fork as an unadulteratedly bad thing. A fork may be painful and unwelcome at the time, but you cannot necessarily know whether it will succeed. Therefore, you and the rest of the project should keep an eye on it, and be prepared not only to absorb features and code where possible, but in the most extreme case to even join the fork if it gains the bulk of the project's mindshare. Of course, you will often be able to predict a fork's likelihood of success by seeing who joins it. If the fork is started by the project's biggest complainer and is joined by a handful of disgruntled developers who weren't behaving constructively anyway, they've essentially solved a problem for you by forking, and you probably don't need to worry about the fork taking momentum away from the original project. But if you see influential and respected developers supporting the fork, you should ask yourself why. Perhaps the project was being overly restrictive, and the best solution is to adopt into the mainline project some or all of the changes contemplated by the fork — in essence, to avoid the fork by becoming it.
All the advice below assumes that you are forking as a last resort. Exhaust all other possibilities before starting a fork. Forking almost always means losing developers, with only an uncertain promise of gaining new ones later. It also means starting out with competition for users' attention: everyone who's about to install the software has to ask themselves: "Hmm, do I want that one or the other one?" Whichever one you are, the situation is messy, because a question has been introduced that wasn't there before. Some people maintain that forks are healthy for the software ecosystem as a whole, by a standard natural selection argument: the fittest will survive, which means that, in the end, everyone gets better software. This may be true from the ecosystem's point of view, but it's not true from the point of view of any individual project. Most forks do not succeed, and most projects are not happy to be forked.
A corollary is that you should not use the threat of a fork as an extremist debating technique — "Do things my way or I'll fork the project!" — because everyone is aware that a fork that fails to attract developers away from the original project is unlikely to survive long. All observers — not just developers, but users and operating system packagers too — will make their own judgement about which side to choose. You should therefore appear extremely reluctant to fork, so that if you finally do it, you can credibly claim it was the only route left.
Do not neglect to take all factors into account in evaluating the potential success of your fork. For example, if many of the developers on a project have the same employer, then even if they are disgruntled and privately in favor of a fork, they are unlikely to say so out loud if they know that their employer is against it. Many free software programmers like to think that having a free license on the code means no one company can dominate development. It is true that the license is, in an ultimate sense, a guarantor of freedom: if others want badly enough to fork the project, and have the resources to do so, they can. But in practice, some projects' development teams are mostly funded by one entity, and there is no point pretending that the entity's support doesn't matter. If it is opposed to the fork, its developers are unlikely to take part, even if they secretly want to.
If, after careful consideration, you still conclude that you must fork, line up support privately first, then announce the fork in a non-hostile tone. Even if you are angry at, or disappointed with, the current maintainers, don't say that in the message. Just dispassionately state what led you to the decision to fork, and that you mean no ill will toward the project from which you're forking. Assuming that you do consider it a fork (as opposed to an emergency preservation of the original project), emphasize that you're forking the code and not the name, and choose a name that does not conflict with the project's name. You can use a name that contains or refers to the original name, as long as it does not open the door to identity confusion. Of course it's fine to explain prominently on the fork's home page that it descends from the original program, and even that it hopes to supplant it. Just don't make users' lives harder by forcing them to untangle an identity dispute.
Finally, you can get things started on the right foot by automatically granting all committers of the original project commit access to the fork, including even those who openly disagreed with the need for a fork. Even if they never use the access, your message is clear: there are disagreements here, but no enemies, and you welcome code contributions from any competent source.
 This is the "pull request" workflow first popularized by GitHub.com (see the section called “Pull Requests”). GitHub's decision to use the term "fork" instead of "clone" to refer to the personal copies in which development is done is largely responsible for the newer, "development fork" sense of "fork".