(apt-get update updates your package lists, not the system.)
# apt-get update
Updates the list of available packages, dependencies, etc, that are available from the repositories as listed in /etc/apt/sources.list
# apt-get upgrade
Goes through the updated list of available packages, checks versions for new revisions of packages that are already installed on the system and presents a list of updated packages for confirmation before pulling and installing the new versions
# apt-get install foo
Checkes the updated list of available packages, checks for any dependencies for foo, and unless foo is the ONLY package, apt-get will present a list of foo and its dependencies that will be downloaded in order for foo to run
Important: If you are running Stable, then there is little to no worry about "breaking" anything if you run "apt-get upgrade". Debian goes through serious work to make sure that Stable packages are exactly that, STABLE. No gratuitous changes go into Stable, only explicit security fixes.
Even if Stable has had a minor version change (I believe Stable is at 6.4 right now), great pains are made to ensure that any and all changes migrate successfully before the new point version is released.
Stable in Debian has nothing to do with software stability, it refers to stability of the packages and packaging system. Fear not! Apt-get upgrade for security today!