How do materials move across cell membranes? compare the different transport mechanisms for moving molecules across the plasma membrane (e. g. is energy required? are the molecules moving down a concentration gradient or against a concentration gradient?

There are 3 methods that materials are transported across cell membranes in basic. Diffusion, osmosis and active transport.

In diffusion, the net movement of fluid is from a region of higher concentration to lower concentration, down the concentration gradient , resulting from the random movement of particles. They do not require extra energy and there will be no more net movement when both sides has a equal concentration.

For osmosis, its the net movement of water molecules only from a region of higher water potential to lower, down the concentration gradient, but must be through a semi permeable membrane (cell membrane in this case). Also, it doesn't require extra energy.

And for active transport, it's the movement of particles against the concentration gradient, by using extra energy provide from for example cellular respiration. It also requires protein carriers.