just so that the process is clear, when your client announces a transaction it sends it to peers which, in turn, relay the transaction to their peers (assuming the transaction is legit) and within seconds the transaction is known to nodes globally.
in most instances, that transaction will be included in the next block, something that can take from a few seconds to tens of minutes.
to monitor the progress, there are some tools. one of these tools is
http://BlockExplorer.com the block explorer site sees a heavy load nowadays and its operator has been trying to make improvements:
http://bitcointalk.org/index.php?topic=64038.0;allan alternate is
http://BlockChain.info that site also recently had some troubles, though looks to be back online and current at the moment.
if your transaction doesn't appear in the blockchain, then the problem is likely on your side. the current bitcoin client is pretty good about making sure no transaction goes out without a fee where one would be needed. but not paying a fee where one is recommended is usually the reason a transaction would be delayed and would not immediately appear in the blockchain.
if your transaction does make it into a block, then the transaction should appear to the recipient.
if the recipient still isn't seeing your transaction, then the problem is likely on the recipient's side. perhaps their client hasn't caught up to the most recent block?
