Passive transport vs active transport sounds like what it is. In passive transport, there is no energy needed to get the molecules across the cell membrane, whereas in active transport, energy is expended getting the molecules across the membrane. Basically in active transport the ATP in your cells gives one of it’s phosphates to the transport proteins in your cell membrane and it lets select molecules in. These molecules are moving through a concentration gradient during active transport(with the help of ATP), from an area of low concentration to an area of high concentration.
Eukaryotes are larger cells with membrane-bounded organelles. Prokarotes are not that large and do not have a nucleus or any memberane-bound organelles.