GPL, also known as copyleft, grants permission to reuse or modify the source code to make derivative works, but if you distribute your program to others, it requires you to license your derivative work under the GPL too. E.g. you have to make the source code available to your users and allow further modification and redistribution. People in the business of making proprietary software (take Microsoft, please) are horrified at the thought that, say, Windows got entangled with the GPL so they'd be forced to release their source code.
LGPL is for library routines. If someone modifies your LGPL library code, the LGPL behaves pretty much like the GPL, but if someone writes software that merely uses your library, the LGPL doesn't impose source release on the program that calls upon your code. This can broaden the potential uses of a library.
AGPL is like the GPL, but the GPL is only triggered if you distribute your derivative work. AGPL broadens this to trigger to activate if you let people use your derivative work over a network. The GNU Affero General Public License is a modified version of the ordinary GNU GPL version 3. It has one added requirement: if you run a modified program on a server and let other users communicate with it there, your server must also allow them to download the source code corresponding to the modified version running there. The purpose of the GNU Affero GPL is to prevent a problem that affects developers of free programs that are often used on servers. Suppose you develop and release a free program under the ordinary GNU GPL. If developer D modifies the program and releases it, the GPL requires him to distribute his version under the GPL too. Thus, if you get a copy of his version, you are free to incorporate some or all of his changes into your own version. But suppose the program is mainly useful on servers. When D modifies the program, he might very likely run it on his own server and never release copies. Then you would never get a copy of the source code of his version, so you would never have the chance to include his changes in your version. You may not like that outcome.
Using the GNU Affero GPL avoids that outcome. If D runs his version on a server that everyone can use, you too can use it. Assuming he has followed the license requirement to let the server's users download the source code of his version, you can do so, and then you can incorporate his changes into your version. (If he hasn't followed it, you have your lawyer complain to him.)
Both the ordinary GNU GPL, version 3, and the GNU Affero GPL have text allowing you to link together modules under these two licenses in one program.
A BSD license allows creation of derivative works, but you could declare your version of the software to be proprietary. E.g. gcc is GPL licensed. You are free to use it's source code to make a better compiler, but when you redistribute your improved compiler, you are obligated to make the improved source code available. If, say, Microsoft liked your version, they could base a version on your code, but the GPL would stick, so Microsoft's version would also be licensed under the GPL. No room for secret undocumented backdoors for the NSA.
Clang (based on LLVM) is BSD licensed. As with gcc you can modify the source if you'd like to improve the compiler. If Microsoft likes your version, they can use it and aren't even obligated to send you a thankyou note. They can make a version of their own based on your work and keep their version proprietary. If they integrate backdoors for the NSA, well, good luck trying to find them when you don't have access to their source code.