The waist is the part of body between the ribs and the pelvis. Waste is the word with many senses mostly relating to things used needlessly or thrown away. Its definitions include (1) to use or expend (something) needlessly, (2) to lose energy, strength, or vitality (usually with away), (3) an act or instance of wasting, and (4) an uncultivated, uninhabited, or devastated area.
The words are pronounced the same, and both have been around a long time, going all the way back to Old English. Aside from that, however, they have nothing in common. Their origins are separate, and they share no definitions. The fact that they are homophones makes them easily confused. As with many homophone pairs, the more common one (waste) is more often used in place of the less common one (waist) than vice-versa. If you need a memory trick to help differentiate them, think of the i in the middle of waist as the word’s waist. The i is in the middle and is narrower than what’s around it, somewhat like the human female waist.