Common apt commands
apt-get install <package> Downloads <package> and all of its dependencies, and installs or upgrades them.apt-get remove [--purge] <package> Removes <package> and any packages that depend on it. --purge specifies that packages should be purged.
apt-get update Updates packages listings from the repo, should be run at least once a week.
apt-get upgrade Upgrades all currently installed packages with those updates available from the repo. should be run once a week.
apt-get dist-upgrade [-u] Similar to apt-get upgrade, except that dist-upgrade will install or remove packages to satisfy dependencies.
apt-cache search <pattern> Searches packages and descriptions for <pattern>.
apt-cache show <package> Shows the full description of <package>.
apt-cache showpkg <package> Shows a lot more detail about <package>, and its relationships to other packages.
man apt Will give you more info on these commands as well as many that are in less common usage.
Common dpkg commands
dpkg -i <package.deb> Installs a package file; one that you downloaded manually, for example.dpkg -c <package.deb> Lists the contents of <package.deb> a .deb file.
dpkg -I <package.deb> Extracts package information from <package.deb> a .deb file.
dpkg -r <package> Removes an installed package named <package>
dpkg -P <package> Purges an installed package named <package>. The difference between remove and purge is that while remove only deletes data and executables, purge also deletes all configuration files in addition.
dpkg -L <package> Gives a listing of all the files installed by <package>. See also dpkg -c for checking the contents of a .deb file.
dpkg -s <package> Shows information on the installed package <package>. See also apt-cache show for viewing package information in the Debian archive and dpkg -I for viewing package information extracted from a .deb file.
dpkg-reconfigure <package> Reconfigures an installed package
man dpkg Will give you more info on these commands as well as many that are in less common usage.