In chemistry, "turnover" refers to the ability of a catalyst to, for lack of a better term, "turn over" after releasing a product molecule. Turnover leads to the regeneration of the "naked" catalyst, which can bind another molecule of starting material and restart the catalytic cycle.
All catalytic cycles operate on the same basic principle: a substrate (or multiple substrates) binds to the catalyst in some way, a chemical transformation of the substrate(s) takes place, and product is released with regeneration of the starting catalyst. A catalytic cycle is analogous to a conveyor belt: substrate goes on the conveyor belt, gets moved to a different position in "chemical space," and is released as the conveyor belt "turns over." Turnover is necessary in order for a process to be catalytic, and achieving catalyst turnover is often the most difficult part of turning a linear reaction into a catalytic cycle. One glaring problem is that the binding affinity of the product for the catalyst must be less than the binding affinity of the substrate for the catalyst. Otherwise, each molecule of catalyst would perform the catalytic cycle only once, then get stuck in the product-bound state. Such a system is missing turnover, and is stoichiometric rather than catalytic. The conveyor belt gets jammed up, in a sense.
Generalizing on this idea, we can say that in order to achieve turnover and avoid logjam, none of the active catalyst species should be too stable. For transition-metal catalysts this is rarely an issue, because transition metals can access a variety of oxidation states (i.e., possess a widely varying number of bonds) and easily bind and release organic molecules. Main-group catalysts are a different story. The common "organic atoms" (C, N, O, H, and the halogens) have oxidation states of widely varying energies, so falling into an unreactive thermodynamic sink along any reaction pathway involving these guys is likely. This problem is often circumvented by the inclusion of a reactive compound in stoichiometric quantities that "kicks up" the catalyst into its most reactive form. Iodine(III), for example, can catalyze C-O bond formation next to a carbonyl group...but to do so requires one full equivalent of mCPBA, in order to reoxidize iodine(I) back to iodine(III) and turn over the catalyst. Without the oxidant, iodine(I) forms upon release of the product and it gets stuck there. A full equivalent of oxidant is required because the catalyst must be reoxidized each time it runs the cycle.
Each run of the catalytic cycle is called a turnover. The number of times each molecule of a catalyst can perform the catalytic cycle before decomposition is called turnover number (TON). Higher TONs are good because they correspond to smaller catalyst loadings, the amount of catalyst required to transform a given amount of substrate. The number of turnovers a catalyst performs in a given time unit is called the turnover frequency (TOF). Higher TOFs are again good because they mean shorter reaction times. The turnover-limiting step of a cycle is the step with the slowest rate constant, which limits how often molecules of the catalyst turn over. You'll often hear people judge the quality of a catalytic cycle by these parameters.