Zeolites are crystalline solids that are interesting in large part because they are porous. Other solids, such as clay, are porous as well, but zeolites are both porous and crystalline (I dont know if they are unique in this regard, but this is certainly a rare characteristic). The porous/crystalline nature of zeolites makes them useful chemical catalysts/filters.
Because zeolites are crystals, their pores are of regular size. Basically a crystal is a solid with a regular, periodic arrangement of atoms. In zeolites, the principal atoms are silicon, aluminum, and oxygen. The atoms are periodically arranged in a manner such that large holes of precise size--we're talking maybe 5 nanometers, but that's huge on a atomic scale--are formed. So you have a network of tunnels or even cavernous "rooms." Of course these things can fill up with water...but more interestingly they can fill up with polymers.
In the petroleum industry, for instance, zeolites are used to trap hydrocarbon chains in crude oil that can fit in the little zeolite tunnels, whereas larger, higher octane rings are too big to fit. So you can filter out the unwanted low-octane fuel and you're left with high-octane good stuff. Chemists would like to use zeolites as catalytic centers--they offer the possibility of molecules meeting eachother in precise locations. I have read about one circumstance in which chemists trapped a catalyst in an internal cavern with branching tunnels through which it couldn't fit. This allowed catalysis to occur with the catalyst isolated from the reactants and products. Doesn't sound all that practical commercially, but it's really remarkable.
Scientists are trying to grow all sorts of new zeolites that aren't found in nature for these purposes. There are about 50 naturally-occuring zeolites. And one final thing. For obvious reasons, zeolites are sometimes called molecular sieves.