Search for packages with “foo” in the name:

Issue the command:

aptitude search foo

Search for installed packages with “foot” in the name

aptitude search '~foo'

Search for packages from the section “foo” that are not installed

aptitude search \!~i~sfoo

For more complex searches and processing, pipelines are your friend:

aptitude search '~i' | grep -ie 'x11\|xorg' | less

NOTE: aptitude search foo is slower than using apt-cache search foo to perform a searc. The output is also a bit different from apt-cache. Use whichever method is preferred.