GRASS requires a workstation running some flavor of UNIX like Solaris, IRIX, Linux, BSD, Mac OS X or cygwin (on Windows). Ideally, you should have at least 500 Mb for data and 32 Mb RAM. The source code package needs around 50 MB uncompressed. The resulting binaries may need between 20 MB and 180 MB depending on your platform. During a full compilation you may need temporarily up to 550MB including the source code.
The Software Download Section of the main GRASS web site contains the latest binaries and source code for all supported platforms. That site also has general directions for installing GRASS manually. However, installation is slightly different on each operating system. Here you can find user-contributed pointers for installing GRASS on specific platforms. In particular, many operating systems have package management utilities that can greatly simplify GRASS installation.
A binary version of GRASS is available from the apt repository. As root type:
apt-get install grass grass-doc
This is the easiest way to install GRASS on Debian. If you choose to install a binary version manually from the main web site, be sure to follow the instructions for making symlinks found as a note to the [6.1 weekly snapshot] release.
There is also the wonderfull DebianGIS project which has a more recent GRASS version with its related packages.
To install from there follow the instructions there
GDF Hannover provides RPM-packages for stable releases prepared to install on SUSE-systems.
You can download them here: 
GRASS Binaries are available from apt/synaptic. From a terminal type:
sudo apt-get install grass grass-doc
or alternatively, search for and install these packages from Synaptic. This is the easy way to get GRASS on your system. Even if you choose to install binaries from another source, you may want to install this version just so that all (most) dependencies are installed as painlessly as possible.
As of 2006-05-21, the latest binary available is GRASS 6.0.1 which is a step version behind the current stable release (GRASS 6.0.2). If you wish to install a more recent binary version, you will need to download the binary package from the GRASS web site and follow the instructions for installing on #Debian. You may find that the effort required to install the binary manually is not much less than compiling the whole package from source. If this is the case, I recommend compiling the development version of GRASS (6.1-cvs) to take advantage of the exiting new capabilities built into 6.1.