Do I need the bitcoin client running in order to receive bitcoin? I am new to this and don't have any bitcoin yet, but I was wondering if I were ever to get any, will I have to have the client running in order to receive them?

Also what would happen if the client it not running at the time a bitcoin is sent to the address?
No, you do not need to have the client running to receive payment on an existing address. Your transaction would go through because bitcoin transactions aren't contained on a local file, but a distributed ledger called the "blockchain."

You don't need to be logged in or anything to receive bitcoin.  What's important is to keep your wallet file containing your bitcoin private keys secret and backed up, as without those, you have no access to the address with your bitcoin.

In fact, a lot of people, fearing their computers being compromised by hackers, actually advocate you store large amounts of bitcoin in a wallet that's generated offline and stored on an offline computer, USB flash drive, or piece of paper.  This practice is called cold storage, and is a very good idea for large amounts of bitcoin you don't want to lose and have no intention to spend quickly.  So, the paranoid strategy would be to never have your wallet online, believe it or not.