What do I do with a TAR.GZ?

A .tar.gz is an uncompiled compressed software package.

Uncompiled - A bunch of code without interpretation.
Compressed - windows equivalent of a zipped file (to simplify)

What do I do with it? First you will need to uncompress this piece of software, Ubuntu has a archive manager for this purpose. That's the easy part. Next if there is any supporting documentation on the install, READ IT FIRST! Then if everything is pretty standard on the install you will need to find the source code and "make" the program.

Making a program: Navigate to the source code for the package src file. In most cases you can just type make to begin compiling there can be other options that appear, you will need the documentation if that is the case. Otherwise on a standard set up the make command will build an exe for you to invoke to install the application. Next make install! More questions ask!

The sudo apt-get install does all this for you if the program you need can be found in the repositories.