Coconut cream is a little like regular cream. It's the rich substance which rises to the surface of freshly made thick coconut milk. Thick coconut milk is the first pressing of grated coconut which has been steeped in boiling water (or sometimes milk), and as the first pressing it is the richest.
The hot coconut milk is refrigerated (it spoils easily), and the thick, fatty layer of cream is spooned off the top after it has cooled. Keep refrigerated and use it soon to top your sticky rice and mango or in your Steamed coconut mussels with kaffir lime (to name a few), as it will not keep for too many days.
Coconut cream is available canned and even powdered for those not prepared to make it at home (I don't know if the powdered is any good, however). Make sure to check the ingredients prior to purchasing. Coconut cream is unsweetened. Cream of the coconut is a different product which is sweetened and used in drinks which typically sport an umbrella.
BlackPawn apparently purchases coconut cream in a block. It's a waxy block which crumbles a bit when cut, similar to cheese, and keeps well. A desired portion is reconstituted with water or added directly to curries and such. BlackPawn says:
I tend to use it in things where it doesn't have to carry the dish though. In say, a Thai curry, I would probably used canned milk. But in most other things, I slice off as much as I need and crumble it into the dish. If the dish is initially dry then it needs to be mixed with a little water but it doesn't take much.
Source: Brennan, Jennifer, The Original Thai Cookbook, Perigee Books, NY, 1981.