Always wash your hands before removing contact lenses. If you are standing in front of a sink, use a clean paper towel to cover the drain where the contact lens might accidentally fall.
To remove soft contact lenses, look upward or sideways while you pull down on your lower eyelid. With a finger, gently maneuver the lens onto the white of your eye. There, you can very gently pinch the lens together with your index finger and thumb and lift it off the eye.
Until you master contact lens removal, you might want to keep your fingernails short to avoid accidentally scratching and damaging your eye.
Rigid contact lenses can be removed by holding out the palm of your hand, bending over, and then opening your eye wide. With one finger of your other hand, pull the skin between your upper and lower eyelid (just outside the lateral aspect of your eye) outward toward your ear with your eye wide open. Then blink. The contact lens should pop right out and into your open palm.