Animal and plant cells are both surounded by a membrane. This membrane is selectively permeable, which means that some chemicals ( such as water and oxygen) can move freely across the membrane, whereas other chemicals ( often large ones like proteins and DNA ) are blocked by the membrane.
The inside of a cell is mainly water, and in any liquid or gas the molecules are moving around at random. This bouncing around means that molecules tend to spread out ( a simple example of this is the smell of perfume spreading across a room).
Diffusion means this gradual spreading out of molecules from areas of high concentration ( the perfume bottle) to areas of lower concentration ( the room ). Another example of diffusion is ink gradually diffusing through a glass of water. The rate of diffusion depends on temperature: as the temperature increases, diffusion speeds up ( ink spreads quicker through hot water than cold water ).
Some chemicals, particularly ones with electrical charge such as sodium ions (Na+) or chloride ions (Cl-), cannot diffuse through the membrane. Instead they move through protein channels ( like little holes in the membrane). This is called facilitated diffusion. Just like regular diffusion, no external energy is required for this.
Osmosis is the diffusion of water through a membrane. The water always moves from a high concentration to a lower concentration. It is very important in your body, because it controls the amount of water in your cells. For example, if you drink water, the water moves from the stomach to the blood, and finally the cells, by osmosis. Obviously you stop drinking when you are no longer thirsty, but animals are not always so lucky.
Some agricultural fairs have competitions for, say, the heaviest pig, and occasionally people who are desperate to win will take their pig and, right before the weighing, put a hose down its throat and fill its stomach with gallons of water to make the pig weigh more. This causes a rush of water by osmosis into the blood of the unfortunate animal, which is extremely painful, and can even kill the pig.
Interestingly this technique of force feeding water to people was one type of torture used in the 13th and 14th centuries by the Inquisition, and also reputedly by the Dutch ( the Dutch tourist board claims that Holland is now a very civilized nation, and tourists are unlikely to be treated in this manner ).
Although forcing people to drink too much water is no longer common, there are cases where athletes drink too much water, and collapse or even die. Yes, that's right, too much water can kill you!. But don't take my word for it, read this article from the New York Times:
When excess water can kill a runner.
Coming back to osmosis, there are three basic types of solution:
Isotonic solutions have the same water concentration on both sides of the cell membrane. Blood is isotonic.
Hypertonic solutions have less water ( and more solute such as salt or sugar ) than a cell. Seawater is hypertonic. If you place an animal or a plant cell in a hypertonic solution, the cell shrinks, because it loses water ( water moves from a higher concentration inside the cell to a lower concentration outside ). So if you get thirsty at the beach drinking seawater makes you even more dehydrated.
Hypotonic solutions have more water than a cell. Tapwater and pure water are hypotonic. A single animal cell ( like a red blood cell) placed in a hypotonic solution will fill up with water and then burst. This is why putting water on a bloodstained piece of clothing makes the stain worse. Plant cells have a cell wall around the outside than stops them from bursting, so a plant cell will swell up in a hypotonic solution, but will not burst.