Monero implements ‘Stelath addresses’, which is one (public) address that you can share with anyone, without enabling spectators to know anything about the transaction history or balance of this address. The Monero addressing system uses 2 private keys: a private viewkey and a private spendkey.

The private spend key pretty much works the same as in Bitcoin: you sign transactions with it.
The private viewkey however is needed to search the blockchain for incoming payments. Only if you have access to that key, you can know a certain transaction output is associated with you Monero address.

When you send a monero transaction, the amount will be split into multiple individual outputs using round numbers (denominations). These individual “bills” will be send to different “one time public keys”. You can’t see which ones are the payment and which ones are the change because the denominations who end up in the same wallet are unlinkable due to the stealth address.

In bitcoin (and most of the other cryptocurrencies) address reuse is often happening, which greatly decreases the pseudonimity of the network. Stealth addresses provide an easy way to protect your privacy and even enhance it.