How can people that had a limb amputated still feel sensation in that limb that isn't even there?

Nerves stimulate areas of the brain that correspond to the part of the body they connect with. The part may be gone, but the brain area isn't and can be stimulated by the remainder of the nerves, other brain areas, or just become slightly active. The brain doesn't know the part is gone and anytime that area is stimulated you feel the body part even if it isn't there due to amputation.

Yes- it's called phantom pains- it's caused by the nerves that are still there giving the false impression of sensation in the missing limb

I think that its because the remaining part of nerves that once ran to the limb carries some impulse to the brain. The brain interpretes this as information comming from the limb and hence the person 'feels' his/her limb.

Yeah, a side effect of amputation is something I *think* is called phantom pains, where they feel pain where the limb used to be.