APT keeps a copy of each downloaded .deb file in the directory /var/cache/apt/archives/. In case of frequent updates, this directory can quickly take a lot of disk space with several versions of each package; you should regularly sort through them. Two commands can be used: apt-get clean entirely empties the directory; apt-get autoclean only removes packages which can no longer be downloaded (because they have disappeared from the Debian mirror) and are therefore clearly useless (the configuration parameter APT::Clean-Installed can prevent the removal of .deb files that are currently installed).

To save up some space via apt-get you can use these commands:

apt-get autoremove
apt-get clean 

autoremove removes unused dependencies, packages which were installed by other packeges but which are no longer needed by your system.

clean just removes all the packages in the apt cache. You can also use autoclean but clean frees up more space.

Another option is to purge unused packages.

aptitude clean --purge-unused

This handy command shows all the packages which are installed on your system because some package recommends it, but they are not actually dependencies of packages:

aptitude search '?and( ?automatic(?reverse-recommends(?installed)), ?not(?automatic(?reverse-depends(?installed))) )' 

can give something like this:

i A apt-xapian-index    -   maintenance and search tools for a Xapian index of Debian packages 
i A exim4               -   metapackage to ease Exim MTA (v4) installation 
i A file                -   Determines file type using "magic" numbers 
i A heirloom-mailx      -   feature-rich BSD mail(1)  

Check the output and if needed remove them all.

WARNING: Ensure that package like libc6, sysv-rc, apt or coreutils are not removed.

aptitude search '?and( ?automatic(?reverse-recommends(?installed)), ?not(?automatic(?reverse-depends(?installed))) )' | awk '{ print $3 }' | xargs dpkg -P

Then we have the all time favorites to see sizes:

df -h
du -h | grep ^[0-9.]*M | sort -rn
du -h | grep ^[0-9.]*G | sort -rn

With these commands I keep my servers clean and lean, but if you do something wrong or remove a package like libc6, sysv-rc, apt ore coreutils you'll have to spend your free evening to fix it up :P.

Note that everything as root. If you get strange errors about permissions then try again as root.