Curry is one word that really should be about 30 words, because there are so many types, some similar, some not. If you want the complete breakdown, the all mighty Wiki is as good of a place to start as anywhere, but it can be broken down in simpler terms.
In general terms, Curry has two main branches - Indian and Asian. In addition, curry can be Wet or Dry.
The first thing to note is that Curry is really a grouping of spices, not a single spice.
Indian Curry is generally dry ground spices, and either cooked dry or cooked in a sauce. Popular spices include turmeric, chilly, ginger, coriander, cloves, cinnamon, peppercorns, cardamom, cumin seeds, garlic, saffron, mace, fenugreek, caraway seeds, and poppy seeds. Onion, garlic, ginger and tomatoes are often then used to make the sauce (sometimes called gravy).
If your going to try Indian Curry recipies, I suggest you buy your curry from Penzey's. Not only great spices, but they will give you a lesson on the different types of curry from India. Note not all Indian curries are hot, although many are.
Thai Curry tends to be wet curry, often coming in a paste, and then often using coconut milk as the carrying agent, more of a soup than just a seasoning. The key ingredients in the majority of Thai curry pastes are mostly wet and fragrant: fresh chilies, lemon grass, galangal, garlic, shallot, kaffir lime,cilantro roots, and shrimp paste (kapee). That last bit is from TempleofThai.com. Often you will see in the store it labeled red, yellow or green curry. Generally you can buy decent Thai curry at the grocery store, but if you want to be adventurous, go to a local asian store. Thai curry almost always spicy when you buy them pre-made, so be careful when adding them to your cooking
I love curry - but it tends to be spicy, so you need to pay attention when cooking. In general terms I find it easier to buy mild Indian curry than Thai curry.