Debian uses a set of tools to manage which software packages are installed. One of these tools is APT, the Advanced Package Tool, which provides an utility called apt-get that lets you install, upgrade or remove packages from your system.

In a nutshell, if you need to install a new package, you usually run:

 apt-get update
 apt-get install package

The first command (apt-get update) forces APT to download the current list of available packages for your machine. After that, apt-get install takes care of downloading, installing and configuring the desired package. It will also take care of any dependencies, if a package needs another package, such as a library, in order to run.

A complete reference for APT can be found in the APT HOWTO.

An alternative to apt-get is aptitude (which you install using apt-get install aptitude, of course). Aptitude lets you interactively browse and search the available packages and their dependencies, but uses a lot more RAM doing so (make sure you have plenty of swap - and patience - available).