What happens when you plug a computer into an ungrounded electrical outlet?
An electrical outlet has a ground connection as a protection device from electric shock. For example, imagine that you plugged your metal (metal conducts electricity very well) kettle into an ungrounded socket. Everything would be OK if the wiring was fine. However, as soon as the cable or wiring becomes damaged then the electricity will pass into the metal kettle - making the kettle 'live' so the next person to touch the kettle will receive a nasty electric shock. Now if the kettle had a ground wire, then the electricity in the kettle would be passed back to the panel box, tripping the breaker and reducing the chances of electric shock. So in the case of your computer, most of the external parts are plastic so the risk of electrocution is minimised as plastic will not conduct electricity very well. However there are many components in your computer that could be damaged, so it is a risk I would not be willing to make as the ground connection will help protect these parts from damage.
Actually, there is a little more to the answer. The ground is still primarily for protection, and plastic or not, a device with a three-pronged plug should be plugged into a grounded outlet. If it didn't need the ground, the plug would be two-pronged. Now in addition to the safety, many of today's sophisticated devices have made the ground an active part of the circuitry, and will be unstable without it. It's hard to say what your particular computer, or any other device, would do, but there's no doubt that it would be better off with it, and may be damaged without it.