This is going to be the long string of numbers also known as a "tx ID". Effectively, it's just a way to look up the specific transaction on the Bitcoin blockchain.
For example, say the address you sent the Bitcoin to is
There are (at the time of this writing) two transactions under that address, one in (green) and one out (red), each for 22BTC. Above each transaction is a long code; for the incoming transaction (green), it's 51eaf04f9dbbc1417dc97e789edd0c37ecda88bac490434e367ea81b71b7b015. This is a tx ID. Using the same sort of block explorer,
you can enter this tx ID and see where the Bitcoin came from and where it went to:
To find the tx ID for your transaction, simply look up the address you sent to, find the amount of Bitcoin you sent (listed) and copy the tx hash displayed for that specific transaction. In the future, many wallets and online services will display this tx ID to you when you send Bitcoin somewhere as confirmation that the funds were sent. I should add, merchants generally won't ask for this; they can just as well look at their wallet/address and see your incoming transaction, even if it's not automated. It is useful to produce if a payment is disputed or can't be found, however.