The "thing" is called a differential, or sometimes, the "rear-end".
If it is leaking where the drive shaft from the engine goes into it, that would indicate the seal is failing.
Many vehicles use 80 or 90 weight gear lube. If you still have the owners manual, it should list the preferred fluid to put in it. Look in the book to see what you need, or call your local Chevy dealer and ask the parts or service guy what you should put in it (he can look it up if you don't have the manual anymore).
If your leak is slight (the casing is wet but not dripping) you can top it off and just keep an eye on it. If its pee-ing grease all over the road, you need to get it fixed before it runs dry and the gears seize up in it.