The block chain records all transactions. So, if your identity is attached to any of your transactions, then your anonymity can be compromised.
For example, suppose you obtain bitcoins from person A and you send bitcoins to person B:
A ----> your wallet ----> B
Person A and Person B know who you are if you have given them identifying information. Person A knows the addresses in your wallet because they sent bitcoins to it. Person B knows the addresses in your wallet because you sent them bitcoins from it. Both Person A and Person B can watch your wallet. If they can determine the identities of addresses associated with transactions from your wallet, they can know where your bitcoins are coming from and where you are sending them to.
Furthermore, law enforcement can gain all of this information with a search warrant or a subpoena.
A coin mixer breaks the chain. Using a mixer once can prevent some tracking. Person A cannot see where you send your bitcoins and Person B cannot see where you got you bitcoins.
A ----> your wallet ??? mixer ??? your other wallet ----> B
Using a mixer twice can make you anonymous by completely isolating your wallet.
A ----> mixer ??? your wallet ??? mixer ----> B
However, there are cases where sending bitcoins from Person A , or to Person B through a mixer may be may not work. One solution is to use an address in your wallet isolated using coin control.
A ----> isolated address ----> mixer ??? your wallet ??? mixer ----> isolated address ----> B
In summary, anonymity is possible with Bitcoin but it is not automatic and it is not easy.
