Osmosis is the way that water moves into and out of plant cells. It generally happens when there are two opposing forces on either side of the cell wall.
The plant cell wall is made up of two membranes, the xylem and the phloem. Both work together like a sieve, allowing some molecules to pass through. When there's a lot of water outside the cell and not very much inside the cell, the pressure from outside forces molecules through the cell wall to the inside, thus equalizing the water pressure. The reverse also holds true. When the pressure inside the cell wall is greater than outside of it, the pressure forces molecules to the outside and the pressure is equalized again.
This pressure against the cell wall is what gives a plant cell its rigidity and shape. When a plant goes too long without water, it will often wilt and droop because its individual cell's walls (xylem and phloem) begin to pull apart from each other and lose their form. As soon as the plant is watered, water rises up the plant from the roots because the pressure is lower inside the plant than outside of it. As soon as the pressure is equalized in each plant cell, the water exchange stops and the plant looks normal again.