what does "make check" do?

make check is a command to a makefile. It does whatever the makefile defines it to do.

It sounds like a little background on makefiles would be good. This is a tutorial that my school uses for a programming course. Here are some good quotes:

Make can be used to automatically execute the many Linux commands that are needed to compile, link, and test a large C++ program. Since these commands will be executed hundreds of times during a program's development, automating these tasks is essential.

According to the GNU Make manual, it performs self tests on the program this makefile builds.

Strictly speaking, it doesn't necessarily do anything.

If a Makefile has a target named check, then make check will "build" that target. It's typically a phony target, meaning that it is a make-scripted command rather than a file named "check" that gets created.

The gnu project advises that all gnu software should include a make check target that runs post-build tests in the build directory, so make check can be used frequently on packages distributed from the FSF. Other projects will sometimes follow this convention as well.