Monero functions somewhat differently than Bitcoin and other cryptocurrencies. While the basic principle of a sender and recipient's account address remains true, what Monero adds to the transaction is a payment ID, which is a hexadecimal string that is 64 characters long. The payment ID serves a dual purpose: It allows the sender to uniquely identify their own outgoing payment, and it also allows the recipient to distinguish the incoming payment from others that they may receive.
Sample payment ID: 8a631acb70e8f96701073cc3b7f33a44ab51be3b40506dd27f4c3ff344d46b32.
Note: While a payment ID is not required to guarantee a successful transaction, sometimes recipients of funds will require the use of a payment ID. For example, the cryptocurrency exchange Poloniex requires that you use a payment ID that they generate in order to guarantee a successful deposit of Monero into your trading account.
Apart from the use of a payment ID, sending Monero is nearly identical to sending Bitcoin. Simply enter your chosen party's account address, select your level of mixin (how many outputs you would like your coins to mix with), and send the funds.
Sample account address: 449Sq9yXRys4phfqoMWGKt7wHnJdcvxKWSENc6im3xXqWWKzJTUe2LnTWQVisy6pTAhFKoTGfSsww7A1L2E3wWVBQhJAgDf.
Receiving Monero is very simple. Simply share the address with whomever you would like to send you money, and await your funds. Please take note that giving the other party a payment ID of your choice may help you in identifying the transaction, though as long as the other party can identify their transaction to you with their own payment ID then either method should work. The exception to this is, as discussed, above, when receiving Monero at a trading account with an exchange such as Poloniex.