Let's start with what the holes in an outlet do. When you look at a normal 120-volt outlet in the United States, there are two vertical slots and then a round hole centered below them. The left slot is slightly larger than the right. The left slot is called "neutral," the right slot is called "hot" and the hole below them is called "ground." The prongs on a plug fit into these slots in the outlet.
If you have read How Batteries Work, you know that electricity must flow in a circuit. In a battery, electricity flows from one terminal of the battery to the other. In a house outlet, power flows from hot to neutral. The appliance you plug into an outlet completes the circuit from the hot slot to the neutral slot, and electricity flows through the appliance to run a motor, heat some coils or whatever. Let's say you plug a light bulb into the outlet. The power will flow from the hot prong, through the filament and back to the neutral prong, creating light in the process.
What if you were to plug a thick strand of wire straight from the hot slot to the neutral slot of an outlet? Unlike an appliance, which limits the amount of electricity that can flow to 60 watts (for a light bulb) or 500 watts (for a toaster), the wire would let an incredible amount of electricity flow through it. Back in the breaker box, the circuit breaker for the outlet would detect this huge surge and it would cut off the flow of electricity. The circuit breaker prevents the wires in the wall or the outlet itself from overheating and starting a fire.
The ground slot and the neutral slot of an outlet are identical. That is, if you go back to the breaker box, you will find that the neutral and ground wires from all of the outlets go to the same place. They all connect to ground (see How Power Distribution Grids Work for details on grounding). Since they both go to the same place, why do you need both?
If you look around your house, what you will find is that just about every appliance with a metal case has a three-prong outlet. This may also include some things, like your computer, that have a metal-encased power supply inside even if the device itself comes in a plastic case. The idea behind grounding is to protect the people who use metal-encased appliances from electric shock. The casing is connected directly to the ground prong.
Let's say that a wire comes loose inside an ungrounded metal case, and the loose wire touches the metal case. If the loose wire is hot, then the metal case is now hot, and anyone who touches it will get a potentially fatal shock. With the case grounded, the electricity from the hot wire flows straight to ground, and this trips the breaker in the breaker box. Now the appliance won't work, but it won't kill you either.
What happens if you cut off the ground prong or use a cheater plug so you can plug a three-prong appliance into a two-prong outlet? Nothing really -- the appliance will still operate. What you have done, however, is disable an important safety feature that protects you from electric shock if a wire comes loose.