This is part of 10998521's Mathematics for the Layman project, which you can read more about on his homenode.
Reductio ad absurdum sounds like fancy Latin, but it's actually a very widely-used method of proof. Say you start with some idea that you want to prove: this is the postulate or conjecture. To prove it, we first of all assume the exact opposite. Then, we have to show that this assumption leads to some contradiction: this could be something obvious like 1 = 3, or more subtle. Then, if the opposite is false, that implies that your original postulate was true. Beautifully logical!
As baffo mentions, among the most famous applications of this is the proof that the square root of two is irrational: it can't be written as one integer divided by another. There is a good explanation of this here, and you can see the steps I described above:
- Assume the opposite: That the square root of 2 is rational
- Show that this leads to a contradiction
- Conclude that it must therefore be irrational.