Here's an open problem that is not hard to understand (high school math is sufficient), but whose solution would be one of the most important advances in mathematics in the last 100 years, easily:

    Let H_n = 1 + 1/2 + 1/3 + ... + 1/n

So, H_1 = 1, H_2 = 1 + 1/2, and so on.

    Let divsum(n) = the sum of the divisors of n

For example, divsum(10) = 1 + 2 + 5 + 10 = 18, and divsum(8) = 1 + 2 + 4 + 8 = 15.

Here's the problem:

    Prove that for n > 1,
      divsum(n) < H_n + exp(H_n)*log(H_n)

It turns out that this is equivalent to the Riemann hypothesis. Their equivalence is shown here: http://arxiv.org/pdf/math/0008177v2