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The GNU Lesser General Public License
The LGPL is often described as "GPL for libraries".
That gives you a rough idea of where to put it in your head, but it misses quite a lot of the detail.
What does it mean in normal English?
Very roughly: the LGPL keeps copyleft requirements around the LGPL-covered library, while allowing that library to be used by programs under other licences, including proprietary programs, if the LGPL conditions are followed.
So can closed-source software use an LGPL library?
Potentially, yes. That is one of the important differences between the LGPL and stronger GPL treatment of combined works.
It does not mean "closed-source software can do absolutely anything with an LGPL library". The LGPL still has conditions.
What if somebody changes the library itself?
Changes to the LGPL-covered library remain subject to the LGPL when distributed.
The copyleft is focused much more closely on the library rather than automatically requiring a larger application using it to use the same licence.
What is all this stuff about linking?
This is the part where the LGPL stops being quite so beginner-friendly.
A program can use a library in different ways. The LGPL contains rules intended to make sure users can modify the LGPL-covered part and, in relevant cases, relink or recombine the application with a modified library.
You will sometimes hear "dynamic linking is fine, static linking isn't". That is far too simplistic to use as your licence policy. The exact licence version and method of distribution matter.
Can I sell a program that uses an LGPL library?
Yes. Commercial software is not banned.
A proprietary application can potentially use an LGPL library while remaining proprietary, provided the LGPL conditions are met.
Why would I choose LGPL instead of MIT?
Suppose you make a library and want lots of programs, including proprietary ones, to be able to use it. But you also want changes to the library itself to remain available under copyleft terms.
That is the sort of goal where LGPL becomes interesting.
MIT would be more permissive: somebody could generally modify your MIT library without the same copyleft requirement on those modifications.
Which LGPL?
Version matters. LGPLv2.1 and LGPLv3 are not just two names for exactly the same text.
Always say which version you're actually using.
An example
You release an audio library under the LGPL. A developer wants to use it in a closed-source game.
That can be possible without forcing the entire game under the LGPL. The developer still needs to comply with the requirements applying to the LGPL-covered library and the way it is combined and distributed.