Was the address you sent to an address set up by the bitcoin.org client? If so, and you installed it recently, you'll need to download the blockchain database up to the point where your transaction occurred in order to see it in the client. (There should be a progress indicator bottom right telling you how far behind you are.) If the address you sent to was pasted OK, and blockchain.info is saying it's confirmed a lot (see below), then you should be OK.

Confirmations are basically new blocks being added to the (most recent) end of the blockchain, the big bitcoin record-keeping database in the cloud. When you send some Bitcoins, the record of your transaction is recorded in (probably) the next block to get mined (which happens roughly every 10-15 minutes). That's 1 confirmation. Each block after that adds on top of it, and adds 1 more confirmation. The more confirmations a transaction has, the more blocks have been added on afterwards. Each block confirms the previous one like a big snake, and each block takes a whole bunch of computing power to add, so after about 6 new blocks following your transaction, it's basically computationally impossible to "undo" your transaction (i.e. find a block that can be used instead of the one with your transaction in).

I think that's right. It may or may not make sense. *Very* happy to be corrected and/or help more on this, as I've started trying to explain it more, and would love to know if I'm being accurate or not...