We already know what diffusion is, and osmosis is simply a special type of diffusion, where only some molecules can move because they can pass through a partially permeable membrane:
The left has a high concentration of water and the right a low concentration. By diffusion they will naturally level out and pass through the membrane. The solute, however will not be able to diffuse, as they can't fit through the membrane.
Therefore we define osmosis as: the net movement of water from a dilute to a concentrated solution, through a partially permeable membrane.
Osmosis plays an important role in maintaining the turgidity of cells. If they are in a solution of distilled water for example, the cell will take on more water molecules without losing the precious nutrients. And become turgid
If, on the other hand, you surround the cell with a highly concentrated glucose solution, there will be a net movement of water molecules away from the cell. Therefore the cell will become floppy and flaccid.