Diffusion and osmosis both refer to the movement of particles across a semipermeable membrane, such as the phospholipid bilayer of the cell membrane. Diffusion and osmosis are both referred to as forms of passive transport because neither require the use of energy (ATP). Instead, they are both powered by a concentration gradient. Diffusion often involves the movement of small particles such as ions. The particles move down a concentration gradient. This means that the particles move from an area of higher concentration to an area of lower concentration.
Osmosis is simply the diffusion of water. Again, the water moves down a concentration gradient (from high centration to low concentration). The water to solute ratio in a cell can cause cells to shrink or burst (lyse) as is shown in the second reference link below.