It depends entirely upon what you want. You have to think about the topic for yourself. As creator of the work, the right to copy the work and prepare derivative works is exclusively your own. By granting a license, you give those rights to third parties. So if you have feelings on the conditions you want to place upon those third parties, it would be a good idea to get that all thought through ahead of time.
Basically, the question you need to ask yourself is this: If someone copies your library and does stuff with it, what do you want to happen?
- If you don't care at all, and want them to just be able to do whatever they please, you can use highly permissive licenses like Creative Commons 0, WTFPL, or even dedicate the work into the public domain.
- If you would like at least an acknowledgement in the documentation, you can use a slightly less permissive license like the various BSD licenses or the MIT license (and the Apache ones are also in there somewhere). They differ in the details, you will need to read them yourself.
- If you want them to release the source code if they change stuff, a copyleft license like GPL or LGPL might be appropriate. The difference essentially being that the LGPL stops itself at the interface boundary of the library. So if they copy your library, change it, and link it into a program, they have to share their copies of your library under the same terms with their customers.
That last point may be worth reiterating: GPL and LGPL only force those that copy your library to use the same licensing terms when passing on their own copies, changed or not. So I for example work at a company that exclusively deals with industry customers. We are under no obligation to pass on even GPLed source code to the general public, only to those industry customers. And they never come asking.
Anyway, for my kernel project, I chose the GPL. Because Linux uses the GPL and that has had positive effects like the OpenWRT project. See, some company (I believe it was LinkSys) used Linux in one of their routers. But they did not use stock Linux, they modified it to run on that hardware. That was before device trees, so the modifications included things like the available devices and their addresses; essentially the Board Support Package. But since Linux was GPLed, LinkSys customers were entitled to a copy of the source code, as modified by LinkSys. LinkSys tried to fight them in court, and lost. And from those files, the OpenWRT project arose.