The main problem with Bitcoin anonymity is that every transaction is publicly logged by design. Anyone can view the flow of Bitcoins from address to address in the blockchain. This data alone cannot be used to identify you because the addresses are just random numbers, but if any of the addresses in a transaction's future or past history can be backtraced to a real identity, it may be possible to find out who owns all the other addresses. Such identity information could be derived from network analysis, surveillance, or a quick google search for the bitcoin address.
If you buy bitcoins using an exchanger, the bitcoins you purchase can be traced back to the exchanger, which may have your real identity information on file, including banking data. All exchanges require the user to scan ID documents, and large transactions must be reported to the proper governmental authority. When you use Bitcoin to pay for goods and services, you will of course need to provide your name and address to the seller for delivery purposes.
A mixing service can be used to mix one's funds with others' or with MIXER's reserve, with the intention of confusing the trail back to the funds' original source. Mixing helps protect privacy, but can also be used for money laundering - mixing illegally obtained funds. After laundering, the funds appear legitimate. Mixing large amounts of money may be illegal, being in violation of anti-structuring laws.