What does Empty Socket mean in Diablo 3?

Answer 1

An empty socket is a socket in which you can insert a gem. There are different kinds of gem, and each do different things depending on where the socket is. For example a Topaz gives an intelligence boost when inserted into torso armour, but a +% to magic find in a helm.

Answer 2

Some items have sockets, into which you can insert gems, granting it a stat boost, etc. Until then, the item will say it has Empty Sockets. Or you may mean the option at the jewelcrafter's, he can remove gems from sockets if you want to replace them.