The Linux kernel, and the overwhelming majority of the code in the GNU operating system (over 90% of a so-called "Linux Operating System" is actually GNU) is written in C.
The one language that you're guaranteed to have already installed with any Linux distribution is the GNU C compiler. If you get the full GCC package set, you have C, C++, Fortran, Ada, a Java compiler for an old version of the JDK, and an Objective C. Only C and C++ are commonly used. The ObjC compiler that isn't much use since most ObjC programs are written using Apple-proprietary class libraries. The GNU Java compiler isn't used much, as it doesn't track the Sun/Oracle standard language. Ada is mostly used by the U.S. Defense Dept. At one time, it looked like Ada might become a rational replacement for Cobol, but that never happened.
For application development ("userland" in Love's answer), you have a choice of pretty much any programming language that's currently in use, but C and C++ are the most common.
Other common languages are (standard) Java and Python. Developers will use other mini-languages in the background. The GNU make tool has it's own language for building projects from source. The shell that runs each terminal session (usually "bash", but others are available) has a very capable scripting language.
Perl is a powerful programming language (with a quirky syntax that programmers tend to either love or detest) used in a lot of admin scripting applications. Because of its power and brevity, it was at one time the primary server-side scripting language for web applications.
All of these are available in free and open source form. The normal way to get something installed is to use the package manager that's used by your particular distribution. A package manager will not only download and install a software package, but will also check for dependencies on or conflicts with other packages before installing. Google for "install packages on
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You can also install any open source project by compiling the source. That's more involved, though, and this answer is already long enough. Almost any free tool will have a web page with instructions on how to compile the source code.