Molecular Resonance is a way to describe the structure of a molecule that could conceivably be constructed in two or more ways.
When you describe a molecule using the standard Structural Formula, you show the atoms and bonds as they should be. In most molecules, there is only one way for the bonds to work out. For example, oxygen gas is O=O and can only be constructed with double bonds between the atoms. Water is H-O-H (that should be bent like an "L"), and can likewise only be constructed in one way -- single bonds between the atoms.
Some molecules, however, can be constructed in more than just one way. For example, nitrogen dioxide (NO2) is a stable molecule. If you try to construct it, you can have either a single bond between the nitrogen and the first oxygen and a double bond between the nitrogen and the second oxygen, or vice versa. Either one is a valid structure, and the true structure is a blend of the two. Thus, both oxygen atoms can be said to have a 1.5 bond with the nitrogen. This molecule is said to have a resonance structure.
Again, the critical point about molecular resonance is that it is a way to graphically describe the bond -- it doesn't physically resonate.
Other molecules that have resonance structures include Ozone (O3) and Benzene (C6H6).