Why does a negative times a negative equal a positive?

Short answer: We make it this way because it makes life easier. For example, consider the distributive property that we have all grown to love:

$$a(b+c) = ab+ac$$

If a negative times a negative were to equal a negative, this property would not hold for negative numbers, which is a bad thing. Let $a=-1, b=1, c=-1$ and $(-1)(-1)=-1$ :

$$-1(1+-1) = (-1)(1)+(-1)(-1)$$

$$-1(0) = -1+-1$$

$$$0$ \ne -2$$

For a longer answer, see this Ask Dr. Math FAQ entry.