This is 40-year-old science (supplemented by a quick dive into wikipedia) so you should probably only use it as a guide. (Note to more up-to-date students than me - I already know I've probably used some out-of date terms.)
Osmosis:
If you have two solutions each side of a semi-permeable membrane, the side with the lower concentration of solute will pass through the membrane more easily, and hence fluid will tend to move from the low concentration side of the membrane to the higher concentration side. The actual movement of the molecules is in fact just brownian motion (qv).
Active Transport:
Similar to osmosis, but we are now considering cells and cellular structures much larger than molecules. The actual movement is caused by cellular energy, rather than brownian motion.
Diffusion:
Is the natural tendency of fluids and gases to even out concentration. If you were to release 1 litre of hydrogen in a typical room, it would be evenly distributed throughout the room within less than a minute, despite the fact that hydrogen is much lighter than the gases that make up air and you would expect it all to float to the ceiling! The same applies to liquids. If you force that diffusion to happen across a membrane or other semi-barrier, you develop "diffusion pressure" (qv) which again seems similar to osmosis or active transport.
