The standard client requires 1 confirmation, except for
change, which can be spent with 0 confirmations (since change is sent from yourself, to yourself, and thus guaranteed to be legitimate). There is a (moderately expensive)
attack against 1-confirmation transactions, though you needn't worry about it unless you're dealing with a relatively large amount of money (and then you just wait for 2 or 3 confirmations). Waiting for 6 confirmations as suggested by others is almost certainly overkill, and there's nothing special about that particular number that makes it "safe" (Why 6? Why is 5 not enough? Why not be extra safe and wait for 7?).
1 or 6. Eligius used to take 24 hours, it was ridiculous.
Eligius (and p2pool) don't actually "send" you coins, instead you get your coins direct from the block reward, which takes 120 confirmations (about 20 hours) to "mature", in case of a blockchain split (which would render the block reward invalid along with all transaction spending it).
