Dipole-dipole and hydrogen bonds are different types of secondary bonding in covalent molecules.
Dipole-dipole interaction are forces between polar molecules but hydrogen bonding is a special kind of dipole-dipole interaction that involves a slightly charged hydrogen atom attaches to one molecule and a slightly charged nitrogen, fluorine or oxygen attached to another molecule.
An example of dipole-dipole force:
Examples of hydrogen bonding force:
Felicia M. pretty much nailed it, I just wanted to offer some analogies.
Dipole-dipole attraction is like having two magnets of opposite polarities that attract each other.
Hydrogen bonding is like if there was a special type of magnet that had a super strong attraction to certain types of other magnets; if coupled with a neodymium magnet the attractive force is much greater than regular magnetic attraction, but if coupled with an iron magnet then the magnetism is normal.
In this example, the neodymium magnet represents charged nitrogen, oxygen, or fluorine (NOF) and the iron is any other dipole.