I sent some coins from my Bitcoin client this morning, but I shut my computer down before any confirmations had gone through. So when I turned my computer back on 70 blocks later or so, I'm seeing the transaction in my Bitcoin client but it's sitting there at 0 confirmations. It's been a couple hours since I turned it back on too. What can I do to get that transaction confirmed?
Probably, the nodes that you were connected to at the time your sent the coins did not relay your transaction and it got lost.
Start your bitcoin client, and leave it running for a while. Transactions that haven't made it into blocks will be rebroadcasted automatically after a while.