Can anyone give me a little information about these different coil types, for example, do they have specific names and what are they used for.

The first two on the left are similar but the second has two distinct windings. The third from the left (in a plastic housing) has two equal windings. The fourth coil has extremely thin wire, I've no idea what it's used for.

 

The last one pictured is a shielded inductor, for applications where sttray magnetic fields are not wanted. The second from last is a power filter for common mode noise. The second from left looks like a current sense transformer from a switching powersupply or it might be another style noise filter . And the first might be used in a speaker cross over. Its big and hefty with lots of large gauge wire, so it probably has a high inductance and can handle large currents.

 

I suggest the 2nd and third are common-mode coupled inductors for typical switchmode applications. The first may also be for a switchmode - as a larger inductance unit for boost or buck, depending on the core.