As Nick already pointed out,
apt-cache search QUERY should list packages containing specific string in their names.
Another useful tool is
apt-file search QUERY will print list of packages containing files with
QUERY in their names. In some cases you know the name of a binary needed (like when you are reading a tutorial and need some command not installed in your OS) but do not know what package it comes with.
dpkg -S QUERY lists installed packages which provide files containing 'QUERY' in their names. In some cases you have a binary or script installed on one machine and want it also on another one, but do not remember where it came from. The output should be smaller than for
apt-file in this case, as it does not display content of packages not installed.
You may filter output of those commands with
egrep, say with
dpkg -S ps | egrep '/bin/ps$' tells us that
ps binary comes with