APT comes in where dpkg stops. The Advanced Packaging Tool, or APT, is a command-line front-end that works with dpkg in Debian (and its derivatives) to manage packages a lot more efficiently. Unlike dpkg where a user has to download a package and its dependencies manually, apt does all the work by itself. It will calculate the dependencies needed for any package, download them, and install everything in order so you don't have to. Since its creation in 1998 and its first release with Debian in 1999, it has been changing and adding features to make itself incredibly useful to all Debian users. A testament to that fact is the number of packaging systems it has been adapted to. Apt can work with rpm, has been ported to Mac OS X, and is also available in OpenSolaris. People are also working on porting apt to the iPhone.
APT is not a single application, but a bunch of applications which together do the work of managing packages on Debian and its variants.
Usage of apt:
apt-get install <package_name> -- Installs a package and its dependencies, if any.
apt-get remove <package_name> -- Removes the specified package.
apt-get update -- Updates the APT package index (A list of all packages available from its repositories)
apt-get upgrade -- Upgrades all the installed packages on the current system.
Repositories: APT relies on directories called repositories to provide it with package files and dependencies. Typically, servers around the world managed by Debian or third-parties are used as repositories. The Debian project maintains a repository of over 25,000 packages which can be downloaded and installed through APT. Other sources of repositories include USB disks, CDROMS and hard-disks.
Frontends: Since APT is a command-line application, it has many front-ends which can be used to simplify the job even more. Two of the most popular frontends available today are Synaptic (graphical) and aptitude (command-line).