I'm a Debian user and--like many--I use apt and its associated tools. If you haven't yet discovered apt here's a brief summary of some of it and some of its tools which can make your package management even more powerful.
##History
I'll start with a brief history lesson. The 'Advanced packaging tool' was introduced in 1998 and first included in a Debian release (2.1 or slink) in 1999. There is no single apt application to speak of, instead it's a series of tools and functions originally designed as a front-end to dpkg -- the software at the heart of Debian's package management (it now also support RPM backends). There are several front-ends to apt itself including: Synaptic, aptitude and Adept but I want to look specifically at the base command-line tools--not that there's anything wrong with the front-ends.
One of the advantages of apt is the repository management. Like most packaging systems, apt uses its repositories to find and retrieve packages. The way in which repositories are organised aids greatly with dependencies--where one package depends upon another in order to work properly. Apt repositories also make recommendations and suggestions of other packages which might prove useful. These are not dependencies as the original package will run without them but sometimes what you might think of as a dependency will be a recommendation because there is a choice involved. Debian policy is not to dictate.
Because of the nature of these tools the user running them needs certain privileges. Usually this would be done with sudo or--if you are confident--as root.
Anyway, on with the tools.
##apt-get update
:~$ sudo apt-get update
Apt makes use of a local cache of packages, detailing the available packages, their version, the files they contain, their dependencies and installation status. Because it's a local cache it needs to be updated with the central lists within remote repositories. This is the equivalent of Synaptic's reload command.
##apt-cache
:~$ sudo apt-cache search gnumeric
:~$ sudo apt-cache search spreadsheet
:~$ sudo apt-cache search show inkscape
:~$ sudo apt-cache depends python
Apt-cache is an interface with the local cache. With it you can search for a package by name or description. For example results for the first two commands above will both include the Gnumeric package but obviously the first one will be more accurate. There are a number of options for apt-cache, search is probably the most popular, show will give detailed information on a package, depends will--as it suggests--show the dependencies for a package.
##apt-get
:~$ sudo apt-get install gnumeric
:~$ sudo apt-get remove inkscape
:~$ sudo apt-get autoremove
Apt-get is the package management side of apt. Using a single command like the first one will retrieve the gnumeric package and all it's dependencies--and all their dependencies--and install them, keeping you informed as it goes. Before this apt-get will ask you to confirm your choice, giving you such information and the download size and eventual storage space required after unpacking. The remove option will not remove dependencies but will prompt you regarding packages that depend upon the one(s) you are trying to remove. The autoremove option will clean up packages automatically installed as a dependency but not removed when the dependent package is removed.
Like apt-cache, there's much more to apt-get than I can cover here. Check out the man files for more on these powerful commands.
##Other useful tools
These tools do not come as part of a default Debian installation so you'll probably need to install them individually.
###apt-show-versions###
:~$ sudo apt-show-versions gnumeric
:~$ sudo apt-show-versions -u
Apt-show-versions is useful for comparing the versions of a package. Generally this will be the installed one and the one in the repository. Using -u switch will give a list of all installed packages for which a newer version is available.
###apt-listbugs
Once installed apt-listbugs is automatically called during an apt-get install process. Before downloading and installing the packages requested (and their dependencies) this will retrieve a list of the critical bugs for them. This is very useful if you are not running a stable version of Debian as you can see which bugs are open, pending or done and look them up before installing.
###apt-mirror
If you are running several Debian installations, you can use this to create a local mirror of some or all of a repository to save your internet bandwidth. This can be useful if you want to ensure all your machines are using particular versions of packages.
apt-proxy
This can also be used to keep bandwidth down for multiple machines but instead it works on a proxy basis, caching repositories updates as clients call on them.
apt-listchanges
This will list the most recent changes (from the changelog) to a package before you install it.
apt-zip
If your machine is not-networked, keeping it up to date with apt can be awkward. apt-zip makes this easier by allowing you to store the packages the machine needs on removable media from another--networked--machine.
Conclusion
Whilst Synaptic and Aptitude have their advantages--for example Aptitude will tidy up orphaned dependencies when removing a package--I think there's a lot to be said for the good old apt tools, particularly if you are on a minimal system or in circumstances where you need to conserve bandwidth etc. I don't claim to be anything like an expert on them and I fully expect others with greater experience to embellish what I've said here. If you've only been using Synaptic give apt a try as well.