Let's provide an example of a proof by contradiction for the somehow dry explanations above:

You might want to prove that there is an infinite amount of
rational numbers (i.e. numbers of the form x/y with x and y being
positive integers) between 0 and 1 (inclusive) *{This is the
hypothesis mentioned above}*. For a proof by contradiction
we assume that there was only a finite amount of rational numbers between 0 and 1
*{negation of the hypothesis}*. But if there
only was a finite amount, we must be able to order the numbers. If we then
counted the ordered rational numbers starting from 1, we must sometime
reach 0 independent of the counting scheme we use *{We
are trying to deduce something}*. So as a counting scheme
we choose to count all the rational numbers 1/x (with x being a positive
integer), which nicely counts an ordered set of rational numbers
between 1 and 0.

Alas, for no x 1/x will reach 0 (no, "infinite" is not
an integer :). Therefore, there is at least one counting scheme which
will never reach 0 *{the contradiction...}*,
which violates the implications we deduced from having only a finite
amount of rational numbers between 0 and 1. But if the implications
are wrong, so has to be the assumption (see implication and
contrapositive). Therefore there is an infinite amount of rational
numbers between 0 and 1 *{... which establishes the
truth of the hypothesis}*. qed

BTW, if you also take real numbers into account,
there is not only an infinite amount but an uncountable amount of
numbers between 0 and 1 (see Real Numbers are Uncountable - proof).