Both of these breaker/outlet types are designed to provide protection for you and your home.
GFCI (Ground Fault Circuit Interrupter) are designed to prevent any excess current from escaping the line and potentially shocking people by turning off the circuit when power goes where it shouldn’t, wiring becomes wet, or damaged.
These are required for many outlets located in kitchens, bathrooms, outdoors and several other locations. If there is ungrounded or leaking current, the GFCI "trips" to prevent the potential for electrical shock. The outlet is then reset with a small button located on the outlet.
GFCI have been required by code since the late 1960's so most homeowners are familiar with these outlets. They are sometimes called GFI.
AFCI (Arc Fault Circuit Interrupter) are designed to protect your home against fires caused by loose wires and other current disruptions that may allow electrical current to arc and start a fire. When these arcs are detected on the circuit, it will "trip" and cut off electricity to the circuit.
Arc faults can be caused by any number of things. Common causes for arc faults are loose wires, wires that have been damaged by nails or animals, damaged or frayed power cords or simply old wiring. Because these arcs often occur in the walls of a home they can be a huge fire risk.
Code now requires that residential areas of your home be updated with AFCI protected circuits during any remodeling or repair work. AFCI protection is available both on a circuit breaker level (inside your electrical panel) and on outlets and receptacles. These look very similar to a GFCI outlet or breaker with a small reset button that tests the current before allowing the electricity to flow again.
Here's a great video that explains the differences: https://www.youtube.com/watch?v=dU91WS94qz4