Many distributions allow you to download packages (such as those ending in .rpm) that contain binaries, basically the program itself. Some programs offer different packages for different distributions, but what is almost always available no matter what application you're trying to get is the source code, provided in a compressed archive ending in .tar.gz or .tar.bz2. Tar is the program that puts the files together in an archive, gzip (gz) or bzip2 (bz2) compress the archive so it takes up less space.
When you install source code, you're doing a few things.
1. Uncompress/extract the archive
tar zxvf filename.tar.gz
tar jxvf filename.tar.bz2
This will take all the files in the package and put them in a directory, usually with the same name as the filename except for the .tar.gz or .tar.bz2 part. The "z" or "j" tells tar whether to uncompress the archive with gzip or bzip2. The "x" tells tar it's extracting the archive, not making one. The "v" means "verbose" and will list the contents of the archive as it's being extracted showing you it's doing something rather than showing nothing until it's done. The "f" means the next argument will be the filename of the archive.
2. Figure out how to install the program
Inside the directory that extracted the archive created, there's usually an INSTALL file (or README file) that explain what other programs you need before you install and the exact steps to follow. However, 95% of them will tell you to do the following:
3. Run the configure script
This runs the configure script (the . is an abbreviation for the current directory, so you're basically typing the full path to the script to run it). This will look at different aspects of your system, make sure you have everything the program requires, and create a "Makefile" that gives the compiler instructions for how to compile the program.
4. Make the program
This will tell the compiler to go through all the source code and create a binary (the program itself) from it. This usually takes the longest of all the steps, you'll see all the compiler commands scrolling across the screen until it's done.
5. Install the program
The previous step compiled the program in the current directory. This step takes all the appropriate files and actually moves them to the various places on your system where they need to be (the binary to bin, the manual files to man, etc.). Since this works with directories other than the current one, it needs to be run as root (type "su" to become root and "exit" to return to your normal user when you're done). The previous commands should be done as a normal user (there's no reason to be root to run them).
That's about it. It seems hard at first, but once you understand what's going on it's really not that complicated and you can use the same steps over and over with everything you install. If you can't find a package right for your distribution you can always install from source, and the source versions are often more updated than the packaged ones.