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Here is an incomplete and badly sorted list of the most useful combinations of apt/dpkg I have been using lately.
Please remember that those command lines which start with a # need root pemissions.

• Install a package from a repository: # apt-get install package
• Install a package in the current directory: # dpkg -i package
• Uninstall a package (keep config files) # apt-get remove package
• Completely uninstall a package # apt-get remove --purge package

• Get the content of a deb package (you must have it somewhere) $ dpkg -c package.deb
• Get description of a deb package (you must have it somewhere) $ dpkg -f package.deb
• Get the description of a package $ apt-cache showpkg package
• Print which packages a given package depends on $ apt-cache depends package
• Print which packages depends on a given package apt-cache rdepends package
• List installed packages $ dpkg -l
• List contents of an installed package $ dpkg -L package
• List description of an installed package $ dpkg -s package

• Look for a package (search only in names) $ apt-cache search -n package
• Look for a package (even in description fields) $ apt-cache package
• Check if a given package is installed $ dpkg -l | grep ^ii | grep package
• Check if an installed package contains a given file $ dpkg -L package | grep file
• Find out which package provided a given file: $ dpkg -S `which file`

Please, feel free to correct this list or add any other useful statement.