To update the local cache from apt repositories

apt-get update

To download and install a package (including dependencies):

apt-get install package-name

To install a single .deb package:

dpkg --install package.deb

To remove a package:

apt-get remove package-name

To find all available packages that match a pattern:

apt-cache search pattern

apt-cache searches the package names and descriptions.

To display package information:

apt-cache show package-name

To show all installed packages:

dpkg --list

To show all files that belong to a package:

dpkg --listfile package-name