AFI – Arc-Fault Circuit Breakers (protection against overloads, short circuits, and parallel arcing)
Breakers with AFI suffix provide standard thermal/magnetic protection, PLUS protection against heat damage caused by parallel arcing in the circuit. A parallel arc occurs when electricity jumps the gap between wires of different voltages. An example of what may cause a parallel arc is a nail protruding through drywall, through the wiring, exposing the bare Hot and Neutral wires. If the wires are close enough together, arcing can occur between the wires. Breakers designed to protect against parallel arcs are called Arc-Fault Circuit Breakers, have a suffix AFI, and have a green test button (early versions had a blue button). An example catalog number is QO120AFI. 

CAFI – Combination Arc-Fault Circuit Breakers (protection against over loads, short circuit, parallel arcing, and series arcing)
Breakers with CAFI suffix provide standard thermal/magnetic protection, PLUS protection against heat damage caused by parallel AND SERIES arcing in the circuit. A series arc occurs when electricity jumps the gap between the strands within the same wire. An example of what may cause a series arc is when an extension cord is damaged because it has been smashed between furniture and the wall, cracking the insulation and exposing strands within one wire. Arcing may occur strand to strand. Breakers designed to protect against parallel and series arcs are called Combination Arc-Fault Circuit Breakers, have a CAFI suffix, and have a white test button. An example catalog number is QO120CAFI.

Which should you use in your home: AFI breakers or CAFI breakers?
This is actually not something you decide on your own. What you use is dictated by your local electric code. Requirements may vary city to city. If you are unsure, check with your local Electrical Inspectors office.