The cells of our bodies are enclosed by semi-permeable membranes. The membranes are made up of phospholipids in two layers. In order to explain how that works, I'll have to explain a little more about phospholipids.

A phospholipid has a polar, hydrophilic "head" and two nonpolar, hydrophobic "tails" that extend from one side of the head, like this:

|  |  ---polar head
 ((   ---two nonpolar tails

Polar things tend to be able to form hydrogen bonds with one another, so the polar water outside the cell forms bonds with the polar heads of the phospholipids. The water also sort of excludes the nonpolar tail section, and it does so more powerfully by way of hydrogen bonds than the bonds formed by the nonpolar tails. The heads of the phospholipids are also attracted to one another, being mutually polar, so the phospholipids have a natural tendency, when suspended in a solution of water, to form two layers which make up a semi-permeable membrane:

Two layer phospholipid membrane:

Exterior of cell
made up of water
 __  __  __  __
|  ||  ||  ||  | ---polar heads form hydrogen bonds with polar water
 --  --  --  --
 ))  ))  ((  ((
 ((  ((  ))  ))
 ))  ))  ((  (( ---excluded by the polar water, nonpolar tails mutually attract
 ((  ((  ))  ))
 nonpolar membrane interior
 ))  ((  ))  ((
 ((  ))  ((  ))
 ))  ((  ))  ((
 ((  ))  ((  ))
 --  --  --  --
|  ||  ||  ||  |
 --  --  --  --

  interior of cell
  made up primarily of water

So anyway, the idea behind the semi-permeable membrane is that it absorbs water into the cell but allows the cell to control the influx to some extent. Life generally exists in water with some kind of solute suspended in it. In our bodies, the most common of these solutes is sugar. The sugar molecule is generally really big, especially when compared to the trim three-atom water molecule, and most other solutes are also fairly unwieldy. The water in the more concentrated side (that is, the more pure water side) will tend to slip over to the other, less concentrated side (the one with all the solute in it) because the pressures on either side tend to force fluid exchange until equilibrium is reached. The solute stays where it is, being too large to slip through the membrane. Water gets between the physical cracks in the membrane in between phospholipids and moves freely inside the nonpolar section of the membrane until it hits something else polar, which most probably would be the water on the other side of the membrane. The whole process is called osmosis, and it's the main reason why cell membranes need to be semi-permeable. Consequentially, though, in too low or high a solute environment the cell will die.

  .  || . .
 .   ||  o. 
 ..  || .   .
.   .|| o  .o
 . . ||  . . 
Water flows from left, less concentrated side to the right, evening out the pressure.

 o=solute, which sticks to a few water molecules
 ||=membrane, which allows the water to move across it, achieving equilibrium

And that's pretty much all I know about it.

The rest is taken from class notes with professor Mark Musch, a very good teacher indeed.
Thanks to vuo, who corrected an error I made.

I am not an expert in this. If I have gotten anything wrong, I'd really like you to tell me. Just /msg sludgeeel.