How anonymous are Bitcoin transactions? I read some sources that say Bitcoin transactions are anonymous, some that say they are semi-anonymous, and some that say they are not anonymous at all.  In what ways are the transactions anonymous and in what ways are they not anonymous?
Technically the appropriate term is "pseudonymous" - imagine that your bitcoin address is like an email address or an online alias: how hard it is to trace to your actions depend largely on what you do with it.

There is, for example, a bitcoin address in my forum signature that would obviously be VERY traceable back to me. On the other hand if I install the client on a separate system which only connects to the internet through TOR and obtain my bitcoins by mining them from a pool like Eligius (no accounts/email necessary, they only ask for a bitcoin address) then it would be very difficult indeed to trace those funds to me.

And of course there are shades of gray in-between. As I said, think of it as being an email address and then consider how your actions with that bitcoin address can be tied to you. It's difficult but possible to make truly anonymous bitcoin transactions.