Riemann surfaces are just a way of visualizing the "multi-valuedness" of complex functions. For example, if z is a complex number, it can be represented in polar coordinates as some radius and some angle from the complex plane. After one complete revolution (360 degrees) you'd expect to get the same result from a function because you should be plugging back in the same point, but that turns out not to be the case, e.g. f(z) = z^(1/2) takes two revolutions to "loop around".
is the simplest I can put it