Osmosis is just a special case of diffusion - where water diffuses through a membrane which has different concentrations on either side.
Permeable means "allows anything to pass through". Membranes in cells allow small molecules (water) to pass through, but prevent bigger ones from passing. This is called "selectively permeable".
Take a look at the following animation;
If a selectively permeable membrane separates the two solutions, water moves through it in both directions at the same time. However, more water leaves a dilute solution (high water concentration) and passes into a more concentrated solution (low water concentration) than enters it. Although the water appears to move across the membrane in one direction, it is in fact moving in both directions but more one way than the other. When the concentration of water is the same on both sides of the membrane, the movement of water will be the same in both directions.
Pure water has the highest water concentration possible. As more salt or sugar is dissolved the water concentration decreases. A concentration gradient is where there is a high water concentration in one area and a lower water concentration in another.
Using these words and information we can now summarise osmosis with its definition.
Osmosis is the movement of water molecules through a selectively permeable membrane from higher water concentration to a lower water concentration - or - down a concentration gradient.
Animal cells are surrounded only by the membrane and may swell up and burst if too much water enters by osmosis. Plant cells have a strong cell wall outside the membrane and this wall prevents them from swelling up too much. They become stiff and hard like a well inflated football. Animal cells just shrivel up when they lose water by osmosis. Plant cells shrink a little, but the tough cell wall keeps its shape when the membrane inside shrinks away from it so the cell becomes limp and floppy like a...