Wikipedia says: "An electrolyte is any substance containing free ions that behaves as an electrically conductive medium".
In human words,(and in the context of hydroxy boosting) an electrolyte makes it easier for electricity to flow through water.
Plain distilled water doesn't really allow current to flow through it. It acts very much like an electrical insulator. We add an electrolyte to distilled water so that electricity can flow more easily. Not only that, but an electrolyte can be used to 'control' the amount of current your electrolyzer uses.
The Theory (if you hate theory, skip down a little)
Ohm's law states that I=V/R, 'I' is current or amps, 'V' is voltage, and 'R' is resistance. With distilled water, the 'R' in the equation is very large, almost approaching infinity. This means that voltage (a relatively small number) divided by a very large number (infinity) equals such a small number that it is basically zero.
When you add an electrolyte to distilled water, it makes the resistance a little smaller, so now when you take V/R, you get a larger 'I' number. If you add more and more, your 'I' get bigger and bigger.
Real life
You've built a booster, its filled with plain distilled water and plugged in to your power supply but not producing any bubbles. Your multimeter(s) says 13.8 Volts and .1 amps. You want to get your amps up to 5 or so so you add a little electrolyte (lets say NaOH). Your amps jump up to 2. You add a little more - 3... a little more... 5.3 amps. You just made your booster pull 5 amps just by changing how much electrolyte is in your water, Amazing!
Problems with 'controlling' your amperage
The amount of amps your booster pulls isn't just defined by how much electrolyte you add. There are other variables that define the 'R' in the equation above. Some of these are:
Example: As most boosters run, they increase in temperature. When the temp goes up, so does the amps. So you started out with 5.3 amps but by the end of your test run your at 20 amps. This can sometimes cause a problem.
The two most commonly used and best electrolytes are KOH and NaOH
KOH - Potassium hydroxide
Pros:
Cons:
NaOH - Sodium hydroxide
Pros:

Cons:
Sodium Bicarbonate - Baking Soda
Pros:
Cons: