Ozonolysis is a way of cleaving carbon-carbon double bonds into two fragments using ozone (O3) as a reagent. The fragments formed are either aldehydes or ketones, depending on the nature of the R groups attached to the double bond, as shown here.
If both R groups on one side of the double bond are alkyl groups, that side of the double bond will become a ketone fragment; if only one R group is an alkyl group and the other R is a hydrogen, that side of the double bond will become an aldehyde fragment.
A quick way of determining the products of ozonolysis is to visually snip the double bond as shown in the next figure, and then to cap both sides with oxygens to make the carbonyl compounds.
A common exam problem is one that asks you to determine the structure of the starting alkene given the products of ozonolysis. If you remember how to determine the products of ozonolysis and then work backward from that reaction (snip off the oxygens, and then smoosh the two pieces together), you get the starting alkene.