Well, no question about distillation is simple because distillation is not simple. But at the risk of over simplification, a pot still is a single container (pot) that is used in batch distillation. The pot is charged with the wash liquid and then heated slowly until lower boiling liquid (alcohol) boils off. The vapor is condensed into liquid and is usually fed to a second pot still for a second distillation. This type of still is widely used in the production of whisky.
The patent still is a name for a column still. It was invented in the 1800's by a Scotsman named Robert Stein. The column still is fed a stream of wash and produces a stream of light product (alcohol) overhead and the heavier product stream as bottoms. This type of still is used to produce large quantities of grain alcohol used in blending.