Imagine someone sprays a deodorant from across the room, eventually it will make its way over to where you are and you can smell it. Diffusion is the spreading out of particles resulting in a net movement from an area of high concentration (where the person sprayed the deodorant), to an area of low concentration (where you are, on the other side of the room). The particles move randomly down a concentration gradient, they are not adventurers on a mission, but tiny particles bumping into each other and eventually reaching dynamic equilibrium where the particles are near-evenly spread out across the room. Some of the substances transported in or out of cells via diffusion are carbon dioxide and oxygen in gaseous exchange, and urea from cells into the blood plasma for excretion in the kidney.
Osmosis is very similar but this is the movement of water molecules. It is defined as: the diffusion of water from a dilute solution to a concentrated solution through a semi-permeable membrane. In the dilute solution there is a high concentration of water, and in the concentrated solution there is a low concentration of water. An example of this is when there is a concentrated solution outside of animal cells, the water moves out of the cell by osmosis down a water potential gradient (basically a water concentration gradient, but measured in water potential) out of the cell, causing the cell to shrivel up and die.