The Riemann Hypothesis is a problem in mathematics which is currently unsolved.
To explain it to you I will have to lay some groundwork.
First: complex numbers, explained. You may have heard the question asked, "what is the square root of minus one?" Well, maths has an answer and we call it i. i multiplied by i equals -1. If the real number line ...-4, -3, -2, -1, 0, 1, 2, 3, 4... is represented as a horizontal line, then the numbers ...-4i, -3i, -2i, -i, 0, i, 2i, 3i, 4i... can be thought of as the vertical axis on this diagram. The whole plane taken together is then called the complex plane. This is a two-dimensional set of numbers.
Every complex number can be represented in the form a+bi. For real numbers, we simply take b=0.
Next: functions. In mathematics, a function is a black box which, when you put a number into it, spits a different number out. A function is represented by a letter - usually "f". If you put a number x into the function you call f, then what f then spits out is written "f(x)".
In most cases there is a convenient way to express f(x) in terms of x. For example, f(x)=x2 is a very simple function. Whatever x you put in, you'll get x2 out. f(1)=1. f(2)=4. f(3)=9. And so on.
You're probably most familiar with real functions, or functions where you put a real number in and always get a real number out. HOWEVER. There's nothing stopping you from putting these weird new complex numbers into a function. For example, if f(x)=x2 and we let x=i, which is the square root of minus one I mentioned above, then you'll get f(i)=-1. That's just the beginning of what's more generally known as complex functions - where you can put any complex number a+bi in and get (potentially) any complex number out.
The Riemann Zeta Function is just such a complex function. "Zeta" is a Greek letter which is written "ζ". For any complex number a+bi, ζ(a+bi) will be another complex number, c+di.
The actual description of the Zeta Function is too boringly complicated to explain here.
Now, a zero of a function is (pretty obviously) a point a+bi where f(a+bi)=0. If f(x)=x2 then the only zero is obviously at 0, where f(0)=0. For the Riemann Zeta Function this is more complicated. It basically has two types of zeros: the "trivial" zeroes, that occur at all negative even integers, that is, -2, -4, -6, -8... and the "nontrivial" zeroes, which are all the OTHER ones.
As far as we know, all the nontrivial zeroes occur at 1/2 + bi for some b. No others have been found in a lot of looking... but are they ALL like that? The Riemann Hypothesis suggests that they are... but nobody has yet been able to prove it.