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/Remove

• 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

Information

• 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

Search files/packages

• 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`
• Find all the packages whose status has changed (i.e. installed/removed) in last #n days:
  $ find /var/lib/dpkg/info/ -name '*.list' -mtime -#n -exec stat {} -c "%y %n" \; | sort -n | awk 'BEGIN{FS="[./]"} {print $1,$7}'

  #n must be a valid positive integer. Minutes can be used in the place of days by replacing -mtime with -mmin.

• Find the last #m packages whose status has changed (i.e. installed/removed) in last #n days:
  $ ls -ltr /var/lib/dpkg/info/*.list | tail -#m

  #m must be a valid positive integer.

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