http://en.wikipedia.org/wiki/MySQL
MySQL is written in C/C++ which are reasonably platform independent - it means a version of MySQL is available for different platforms (versions of operating systems). Binaries are build for different platforms, and you need to download binary version for the platform you have. Windows binary would not work on BSD Unix, etc.
You can see available platforms here: http://dev.mysql.com/downloads/mysql/
Likely with some skills you can build MySQL from sources on many other platforms. Not sure what exactly you want.
If you download source version, you need to have build tools. For example on Windows, you need to install C and C++ compilers and other tools as necessary.
Or you can just download both binaries and sources, and read source for learning experience.
Wikipedia article above has links explaining many more terms, to clear many more of your doubts. :-)
Edit:
C and C++ are considered "platform-independent": they are part of Gnu compiler collection : which provides compilers for many languages ("The GNU Compiler Collection includes front ends for C, C++, Objective-C, Fortran, Java, Ada, and Go, as well as libraries for these languages") for many platforms/operating systems. And of course they power all distributions of Linux. See http://en.wikipedia.org/wiki/GNU_Compiler_Collection and http://en.wikipedia.org/wiki/GNU_Project