Bitcoin is generally described as an anonymous currency. The major reason being the possibility to receive and send bitcoins without mentioning any personal identity or information. But, as a matter of fact, absolute anonymity is also not possible. It can be tracked if need be.
The correct way to mention Bitcoin is to say that it is pseudonymous. Sending and receiving is similar to writing under a pseudonym. If the pseudonym of the author ever gets linked to the identity, then it is always possible to connect the two easily. Relating it to Bitcoin, the address acts as a pseudonym. If the address is linked to your indentity, it can be easy to figure out your identity.
In the actual Satoshi whitepaper, it was suggested that the user should use a different address every time for a send or receive transaction. This can help to keep the identity hidden and no one can trace the identity by linking the address and the identity.
Multi-input Transactions occur when you receive payments to your wallet to different addresses, but later you send the payment out of your own wallet which pulls the bitcoin from multiple addresses. This transaction will include multiple addresses as inputs, which clearly prove that they are in the same wallet and belong to the same person or entity. Now, if the identity is ever linked to the addresses, then it cannot be possible to maintain the identity hidden at all.
In case you intend to increase the anonymity, you can prefer using multiple wallets. It can be like having different identities.
A mixing service is a firm or entity that provides a service to trade out the bitcoins you hold for the other ones that have a different history. In order to conduct this, they take your bitcoins and mix it with a huge pot of bitcoins from various other users. Then they trade out randomly picked bitcoins which makes it impossible to figure out which input and output is connected.
Also, an option of Web Hosted wallets can be opted for. It can be used as an effective way to mask the original owner of the Bitcoin. Numerous web-based wallet services will mix the bitcoins in their service together and then providing randomly to conceal the identity. Most of the web wallet service providers maintain a record of incoming and outgoing coins. Thus the anonymity factor is dependent on the Web wallet service provider.