When the environment outside of the cell membrane has an equal water concentration as the inside the cell there is no visible change. Both animal and plant cells respond the same in isotonic solutions because there is no net movement of molecules.
When the environment outside of a cell membrane has a higher water concentration the water will flow into the cell to balance the solute to solvent concentration. Animal cells begin to swell when placed in a hypertonic solution, and if it swells to much it may burst. Contrary to the reaction of an animal cell plant cells simply become firmer and do not burst thanks to their cell wall.
Osmosis is the process in which a fluid such as water passes through a semi-permeable membrane, from areas of high solvent concentration to areas of low solvent concentration.
When the environment outside a cell membrane has a lower water (solvent) concentration, the water will flow out of the cell to balance the concentration inside and outside of the cell. In the case of animal cells, they shrink in hypotonic solutions, whereas a plant cell would simply become "soft" or "limp".