@grue@lemmy.world 's explanation is spot on. However, just to practice summaries, I’ll write anyway.

Here are a few terms I’ll use:

• Project A
• Project B

The GPL family of licenses are all copyleft. This means that any code used from Project A in Project B, for instance, makes the latter have the same license, along with not being able to make either Project A or B proprietary*. Furthermore, with copyleft licensed software, you have to give the corresponding source code, along with the compiled binary.

The difference between the v2 and v3 versions of GPL is that, with the former, Tivoization was possible. Tivoization makes you only able to use the “official” version of the software, rendering it proprietary. Now, here is the comparison between the GPL variants:

• GPL: The first of the kind, not much to say here.
• LGPL: It allows you to link (as in libraries) code from Project A to B, without making Project B use the license, too
• AGPL: You’ve certainly heard of SaaS, which allows you to serve websites licensed under the GPL, for example, without giving the corresponding source code. The AGPL solves this.

*That is, if there are more contributors to Project A that have not given up their authorship. Also, the version of Project A that used the GPLv3 before can still be used under the same license.