Debian's APT system is made up of many individual tools that allow you to do fun things with your packages and package database. Here's a list of them; this is not meant to be a command reference, so command-line flags, etc, are not discussed. This is simply meant to be a full list of available commands, with general descriptions, so that a reader can know what's out there and what he might want to research/install...

dpkg

This is the basic package utility, that handles installing packages, removing them, and getting information from them. All the other tools are frontends for dpkg, and many users rarely use it directly.

apt-get

Every Debian user knows this one; it's the basic tool that's used to organize the installation of packages (dpkg does the actual installing, apt-get handles dependencies, etc). Apt-get itself is text-mode, although it may ask installation questions in a X11 window depending on system configuration.

apt-cache

This command is used to search the contents of your local package database. Searches with apt-cache are different from searches with dpkg in that an apt-cache search will cover the entire contents whatever is in your sources list (as opposed to dpkg searches, which only search the contents of installed packages). Text-mode.

dselect

An ncurses-based frontend for apt-get that handles package dependencies, suggestions, recommendations, etc. Use of this program is recommended for not necessarily "advanced" users, but for people who are willing to wade through a very terse and sometimes confusing interface. Many people have blamed this tool for making sweeping changes to their package installations that they didn't intend to make. Approach with caution. :)

aptitude

Another ncurses-based frontend for apt-get that's a lot friendlier. Recommended as a precursor to, or replacement for, dselect.

synaptic

Originally designed for !SuSe? Linux, there is a Debian-adapted version of this X11-based apt-get frontend that works really well for what I call "package shopping"...going through the full list of what's available. I personally had problems doing the actual installing with synaptic, so personally I recommend running synaptic for package shopping, then opening an xterm and running apt-get at the command line to actually install...but that was a while ago and it's quite possible those issues are resolved now.

apt-file

Search for a filename installed by any package in the contents of sources.list.

apt-listchanges

Display the relevant changelog entries whenever a package is upgraded.

apt-build

Download the source package for something, build it on the fly, and install the newly built binary package. This normally isn't necessary, since Debian's packages will already be built against the libraries you have (since Debian, unlike other distributions, is a coherent package set rather than just an FTP site full of unrelated packages), but with apt-build you can optionally apply patches to the source tree after downloading but before building. The command "apt-build world" does what you'd expect it to do...I'm afraid of this and haven't tried it myself. :)

apt-cdrom

Add a CD-ROM with packages on it to your sources.list.

apt-rdepends

Generates recursive dependency trees for a package.

apt-spy

Generates a sources.list file from the current list of mirrors, after testing them for latency/bandwidth.

apt-show-versions

Shows all available versions of a package (allows wildcards), or all packages, and displays upgrade information where applicable.

dpkg-reconfigure

Allows you to reconfigure package preferences (the same questions that are asked when the package is installed).

apt-show-source

Variant of "apt-cache show" that shows information for the source version of the package named, rather than the binary version.

apt-listbugs

Shows critical bugs in named packages. Can be run anytime, but specifically intended to help users that track the "unstable" distribution from upgrading to a package that catastrophically breaks the system.

apt-zip

Allows non-networked machines to benefit from apt by exporting a list of packages selected for install, then later installing those packages from removable media after another machine has downloaded them.

apt-config

This one doesn't really count, as it's not meant to be used directly...this program is used by the other apt utilities to provide a standard interface to the apt configuration settings.

apt-ftparchive

Generates, from a directory of packages, the "Packages" and "Sources" control files used for an apt repository.

apt-move

Moves a collection of Debian packages into a proper "pool" directory hierarchy as used by a package repository.

apt-sortpkgs

Recreates Sources and Packages files with the records sorted and within each record, the fields sorted.

apt-extracttemplates

Extracts templates and install scripts from packages.

apt-proxy

Specialized HTTP proxy server made to proxy access to one or more apt repositories. This is very useful on a LAN, where the entire network can get packages from a central machine that only downloads from the mirrors once.

apt-proxy-import

Imports packages into a cache used by apt-proxy.

apt-setup

Sets up the sources.list file (this gets run during the initial OS install, for example)

apt-cacher

Alternative to apt-proxy

auto-apt

From the package description: auto-apt checks the file access of programs running within its environments, and if a program tries to access a file known to belong in an uninstalled package, auto-apt will install that package using apt-get.