The compiler is a program that takes source code (the code written in the C programming language in the case of Asterisk) and turns it into a program that can be executed. While any C compiler should be able to compile the Asterisk code, we strongly recommend that you use the GCC compiler. Not only is it the most popular free C compiler on Linux and Unix systems, but it's also the compiler that the Asterisk developers are using.
If the GCC compiler isn't already installed on your machine, simply use appropriate package management system on your machine to install it. You'll also want to install the C++ portion of GCC as well, as certain Asterisk modules will use it.