Although there are many different free software licenses available, in the important respects they all say the same things: that anyone can see and use the code, that anyone can modify the code, that anyone can redistribute it both in original and modified form, and that the copyright holders and authors provide no warranties whatsoever (avoiding liability is especially important given that downstream recipients might run modified versions without even knowing it). The differences between licenses boil down to a few oft-recurring issues:
The non-reciprocal (non-copyleft) free licenses allow the covered code to be used in proprietary programs. This does not affect the licensing terms of the proprietary program: it is still as proprietary as ever, it just happens to contain some code from a non-proprietary source. The Apache License, X Consortium License, BSD-style license, and the MIT-style license are all examples of proprietary-compatible licenses.
Most of the commonly-used non-reciprocal free licenses are compatible with each other, meaning that code under one license can be combined with code under another, and the result distributed under either license without violating the terms of the other. Some of them are also compatible with some of the copyleft licenses, meaning that a work comprised of code under the non-reciprocal license and code under the copyleft license can be distributed as a combined work under the copyleft license (since that's the license that places more conditions), with the original code in each case remaining under its original license. Typically these compatibility issues come up between some non-reciprocal license and the GNU General Public License. This topic is discussed in more detail in the section called “The GPL and License Compatibility”.
Some free licenses stipulate that any use of the covered code be accompanied by a notice, whose placement and display is usually specified, giving credit to the authors or copyright holders of the code. These licenses are often still proprietary-compatible: they do not necessarily demand that the derivative work be free, itself, merely that credit be given for its free parts.
This is a type of attribution requirement. Trademark-protecting licenses specify that the name of the original software (or its copyright holders, or their institution, etc.) may not be used to identify derivative works, at least not without prior written permission. This restriction can be implemented purely via trademark law anyway, whether or not it is also stipulated in the copyright license, so such clauses are somewhat legally redundant — in effect, they amplify a trademark infringement into a copyright infringement as well.
Although attribution requirements insist that a certain name be used, while trademark protections insist that it not be used, they are both expressions of the same concept: that the original code's reputation be preserved, and not tarnished by associations beyond its control.
Certain licenses (e.g., the GNU General Public License version 3, the Apache License version 2, the Mozilla Public License 2.0, and a few others) contain language designed to prevent people from using patent law to take away the rights granted under copyright law by the licenses. They require contributors to grant patent licenses along with their contribution, covering any patents licenseable by the contributor that would be infringed by their contribution (or by the incorporation of their contribution into the work as a whole). Then they go further: if someone using software under the license initiates patent litigation against another party, claiming that the covered work infringes, the initiator automatically loses all the patent grants otherwise provided for that work by the license, and in the case of the GPL-3.0 loses their right to distribute under the license altogether.
Most of these stipulations are not mutually exclusive, and some licenses include several. The common thread among them is that they place certain easily satisfiable demands on the recipient in exchange for the recipient's right to use the code under the freedoms granted by the license.
 Or its variant, the GNU Affero GPL (see the section called “The GNU Affero GPL: A Version of the GNU GPL for Server-Side Code”).