Nicolas' answer is only partially correct. For renting it's the move-in date, but for buying it's not the closing date, although it's the legal event equivalent to the contract-exchange date in the UK, Australia and some other countries.
You would ask your own real-estate agent "What is the closing date?" or "When do we close?", that is the day all the paperwork for the sale is completed and you legally own the new house. The sales contract usually stipulates "close + x" where x can be 1 day and up to 3 days, that is the period you and the seller agreed ahead of time to allow the seller to completely move out, you get the keys thus "access" to your new house after this period. The specific terminology is "getting the keys", You ask your real-estate agent "When do I get the keys?" and she would answer for example "Close plus one". Note that this is also written in the sales contract so it's a legal requirement, the real-estate agent, even if she has the keys in her possession, is not allowed to hand over the keys to you, and legally you cannot enter the house until then. (You could, by prior agreement with the seller, have a few things, bulky furniture for example, moved in during this x-plus period). This is the regulations in California by the way, other states may be slightly different.
PS. Hi Mikkel, it sounds like you're more interested in the terminology for the UK and Commonwealth countries, but I just remembered another term used in the U.S. It's "possession date". You ask your agent "When is my possession date?" or "When do I take possession?", and she would answer "Same as Close date" or "Close plus one" as I mentioned above.