how in the world is diffusion, felicitated diffusion and active transport related?

Answer 1

Simple diffusion is a process that takes place due solely to the concentration gradient present across a barrier. in this case I assume your referring to a cell membrane. Here molecules cross the membrane without any "assistance"

Facilitated Diffusion is what happens when the molecules (usually proteins) cross the membrane through specific channels, which facilitate the crossing.

Active Transport can go against the concentration gradient, with the active help of transport proteins.

all three of these are methods of transport for molecules across membranes, and are necessary for biological function.

Answer 2

These are all ways for materials to pass through the plasma membrane - entering or leaving the cell. The first two are passive, and active transport requires energy.

1. Diffusion - simple net movement of molecules from an area of higher concentration to an area of lower concentration.
2. Facilitated diffusion - still moves molecules from high to low, but the molecules pass through special proteins that are always changing shape.
3. Active transport - moves molecules from LOWer concentration to HIGHer concentration by trapping the molecules in special transport proteins on the low side and flinging them out on the other side when the cell uses ATP energy to change the shape of the protein.