In order to compile/link a C program, you just need a compiler/linker combination. There are a few options out there (gcc from GNU,vc from Microsoft, and others). Programs that have "text highlighting" and other similar syntax and search tools that help a programmer are called Integrated Development Environments (IDEs). These include code blocks, netbeans, and of course, visual studio. If you don't want too much glitter for your coding, you can simply use a text editor that has some syntax highlighting (such as what this tool you're asking about is, or other text editors such as gEdit).
Depending on what you choose to use, the compilation is going to be a bit different. If you use an IDE, you usually just have to push a button to compile, link, and run. If you use a text editor, you have to compile/link outside of the editor (through command line), then run the code through the command line or by double-clicking on it in an explorer window.
Sounds like you just need a starter guide to C (using gcc in this case):