Both osmosis and diffusion are ways by which particles move.  Each are based around the idea of concentration. We can describe the concentration as being either high or low. The diagram below shows an area of a high concentration surrounded by a substance of very low concentration:

Diffusion is the movement of particles from a region of a high concentration to a region of a low concentration until they reach an equilibrium (equal concentrations). So if we take the diagram above and redo it to what it would look like after diffusion you would get something like this: 

In the above diagram all of the particles are spread out and all at about the same concentration. This will not happen instantly. A real life example of diffusion is found in tea making. When you put a tea bag (high tea concentration) in boiling water (low tea concentration) the tea moves out the tea bag into the water by diffusion.       

Osmosis[i] is much like diffusions as it is concerned with high and low concentrations except osmosis is all about water. Osmosis is defined as the movement of water from a region of high concentration to a region of low concentration down a concentration gradient.         

                                                                                                                                      If you dissolve something (say salt) in water the water becomes less concentrated. So if you placed a cell of high water concentration in the solution you would find that water had moved from the cell into the solution until they were equal in concentration.

[i] Years ago in a class on the topic someone asked: “Osmosis wasn't he the guy from the bible?”