A wallet is software used to access the block chain. It does not contain any coins. It just holds your keys. You can use the same keys on as many different computers as you want (just like you can make copies of your car's key and put them in any pocket you want), but you shouldn't because that might confuse you and result in lost coins.
Yes okay, but does the new client receive the new keys? So if I give out my address, then back up my wallet and load it on a new computer, does this new computer receive the keys when someone sends coins to the old address?
The terminology can be confusing. "bitcoin address" and "public key" are the same thing. The wallet contains your keys, both public/address and private, so when you load it on a new computer you are copying the keys and addresses to the wallet on the new computer.
Okay but I transferred my lite coin wallet to a new computer, and when I loaded it up on the new computer there was no address associated with it. When Someone sent coins to the original address it showed up on the original litecoin client, but not the one running on the new computer.