If I put a red blood cell into pure water it will swell and burst, why is this?

Because of osmotic pressure. The cell has electrolytes dissolved in it, fluid (H2O) travels from outside the cell, through the membrane of the cell and into the cell. This continues and the water is TRYING to dilute the solution (technically, its the osmotic pressure doing the pushing or pulling depending upon how you want to look at it) and finally there is enough fluid inside the membrane that the cell wall can't hold it any longer and the cell bursts.

http://en.wikipedia.org/wiki/Osmotic_pressure

When a red blood cell is placed in pure water, it undergoes a process called osmosis. Osmosis is the movement of water across a selectively permeable membrane, such as the cell membrane, from an area of lower solute concentration to an area of higher solute concentration.

In this case, the red blood cell contains solutes such as ions and proteins, while pure water does not contain these solutes. Since the solute concentration inside the cell is higher than the solute concentration in pure water, water molecules diffuse into the cell via osmosis in an attempt to equalize the concentrations on both sides of the cell membrane.

As water enters the cell, the volume of the cell increases, causing it to swell or enlarge. However, red blood cells are flexible and can withstand a certain degree of swelling. But if the water continues to flow into the cell, the intracellular pressure increases, and eventually, the cell reaches its maximum volume capacity. At this point, the pressure becomes too much for the cell membrane to handle, resulting in the cell bursting or undergoing lysis.

So, to summarize, when a red blood cell is placed in pure water, water molecules enter the cell through osmosis, causing it to swell and eventually burst due to the increase in intracellular pressure.