I have grouped these two topics together, although they serve different purposes.
RF coils are the "antenna" of the MRI system that broadcasts the RF signal to the patient and/or receives the return signal. RF coils can be receive-only, in which case the body coil is used as a transmitter; or transmit and receive (transceiver).
Surface coils are the simplest design of coil. They are simply a loop of wire, either circular or rectangular, that is placed over the region of interest. The depth of the image of a surface coil is generally limited to about one radius. Surface coils are commonly used for spines, shoulders, TMJ's, and other relatively small body parts.
Paired saddle coils are commonly used for imaging of the knee. These coils provide better homogeneity of the RF in the area of interest and are used as volume coils, unlike surface coils. Paired saddle coils are also used for the x and y gradient coils. By running current in opposite directions in the two halves of the gradient coil, the magnetic field is made stronger near one and weaker near the other.
The Helmholtz pair coils consist of two circular coils parallel to each other. They are used as z gradient coils in MRI scanners. They are also used occasionally as RF coils for pelvis imaging and cervical spine imaging.
The bird cage coil provides the best RF homogeneity of all the RF coils. It has the appearance of a bird cage; hence, its name. This coil is commonly used as a transceiver coil for imaging of the head. This type of coil is also used occasionally for imaging of the extremities, such as the knees.