Privacy is an important property of any functioning currency.
With bitcoin you already has several options to be private on the network. The problem however is that some of these solutions are not really safe, others aren’t very user friendly. But the biggest risk is ignored very often: by using a payment network that isn’t private by default, your identity may be associated with (some) BTC addresses when you transact with people who are not concealing their identity on the blockchain. Check here for more information on Bitcoin privacy issues.
Monero enforces the use a special kind of addressing system called “stealth addresses”. By using this obfuscation method by default, it’s much harder to do blockchain analysis. More info on how the stealth addresses work can be found here.
Although it is still possible to group certain inputs together and attach them to identities in some edge cases. The Monero Research Lab wrote a paper on this subject and can be found here: MRL-0004: Improving Obfuscation in the CryptoNote Protocol. Most of the attacks on the privacy will probably be solved once “RingCT” will be implemented in Monero. You can read more on this here.