Osmosis is essentially the diffusion of water. Diffusion is a process that happens naturally when a certain fluid (liquid or gas) is in a higher concentration in one place than it is in another. The fluid tends to move from the area of higher concentration to the area of lower concentration. This difference in concentration is known as "the concentration gradient".
Here's an example most of us can relate to. If you've ever been visited by a grandmother or elderly aunt, I'm sure you know what it's like to get a big whiff of strong perfume. Even if you don't stand near the woman, you can still smell the perfume. This is because the perfume is in high concentration in the area surrounding her body, but the perfume is in lower concentration around you, so the perfume molecule naturally move through the air towards you.
This happens with water too, and can have a very drastic impact on cells. When diffusion happens in water, it's called osmosis.
If a cell is in a salty solution, the cell is said to be "hypertonic" to the solution. Hypertonic means that the concentration of water is higher INSIDE the cell than it is OUTSIDE the cell. So which would you suspect to happen: water move into the cell, or leave the cell? If you said "leave the cell", you would be correct. Because the outside environment is salty, there's a lower concentration of water molecules, so the water inside the cell would tend to leave the cell. This can cause a serious problem for both animal and plant cells. If too much water leaves a cell, it can shrivel up and die.
However, if the cell is placed in pure water, the process of osmosis will go in the other direction. Because a cell has other molecules besides water inside it, it would have a lower concentration of water when placed in a pure water solution. So, remembering what we just learned about diffusion, we would know that water would move into the cell. We would say that the cell is "hypotonic" to the solution. This means that the concentration of water is higher OUTSIDE the cell than it is INSIDE the cell.
Now an interesting thing happens here. For animal cells, this is also a potentially deadly situation. If too much water enters the cell, the cell can swell and the cell membrane can actually explode, killing the cell. Some unicellular organisms, such as a paramecium or a euglena, have a special structure to prevent this from happening. They have a small hole called a contractile vacuole, which will pump out excess water, preventing the cell from exploding.
In plants, however, it's different. This is actually the preferred situations for plants. Because plant cells are protected by a tough cell wall, they will not explode in a hypotonic situation. It causes the plant to become strong and stand upright. This is why plants will perk up after they are watered.
So, just remember.
HYPERtonic = greater concentration INSIDE the cell.
HYPOtonic = greater concentration OUTSIDE the cell.
I know they're similar words, but that tiny difference means a very different thing.