What molecules help substances move through the cell membranes during active transport and facilitated diffusion?
The majority of the cell membrane is composed of a double layer (aka bilayer) of phospholipid molecules, which are responsible for maintaining the separation between the cell's contents and the exterior environment. This is also called semi-permeability, or selective permeability, because things can be transported across the membrane, but only if they meet certain criteria.
Some textbooks and descriptions of the semipermeable nature of the membrane make it sound as if materials simply pass straight through the membrane, perhaps by working their way in between the phospholipids and then coming out the other side. While this is possible, it's inefficient and contradicts the idea of using the membrane to regulate transport. Instead, proteins that are embedded in the membrane are used to transport materials across it.
It bears reminding that proteins are typically large and complex, especially in comparison to a phospholipid molecule, and are therefore capable of more complex functions and behaviors than a phospholipid. Typically a protein embedded in the membrane will have some sort of interior opening, like a tube or channel, through which certain molecules can pass. The molecules can only pass if they fit the physical size and electromagnetic charge of the surrounding protein. In this way, the protein acts like a combination of a door and a guard or bouncer, ensuring that only the things which fit through the "door" are allowed to pass through.
The only real difference between facilitated diffusion and active transport is that active transport requires energy in order to take place. This is typically because the protein is working against a concentration gradient, attempting to pack even more molecules into an already highly concentration area, whereas diffusion involves moving from high to low concentration, and therefore requires no energy.