Reading the Riemann Hypothesis Writeups and the Riemann Zeta Function and trying to understand them will help understanding what this is, though the math is rather heavy.
The generalized Riemann hypothesis was probably formulated for the first time by Piltz in 1884. Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime numbers.

The formal statement of the hypothesis follows (sorry if you don't): A Dirichlet character is a completely multiplicative arithmetic function χ such that there exists a positive integer k with χ(n + k) = χ(n) for all n and χ(n) = 0 whenever gcd(n, k) > 1. If such a character is given, we define the corresponding Dirichlet L-function by L(X,s)= Σ(n=1 to ∞) (X(n)/N^{s}) for every complex number s with real part > 1. By analytic continuation, this function can be extended to a function defined on the whole complex plane. The generalized Riemann hypothesis asserts that for every Dirichlet character χ and every complex number s with L(χ,s) = 0: if the real part of s is between 0 and 1, then it is actually 1/2. The case χ(n) = 1 for all n yields the ordinary Riemann hypothesis.

I'll try to go through that slower:
There's a Riemann Zeta Function, ζ(s). If you replace a bunch of the constants in the function with variables, and put it onto the complex plane (including imaginary numbers), you get a more complex, but more general function (The extended The Dirichlet L-function) that describes a whole lot of other functions, including the Riemann Zeta Function. Instead of just proving that the zero's of the Riemann Zeta Function all lie on σ = 1/2, it would be nice to show that a special case of this Generalized Riemann Zeta function (that's the The Dirichlet L-function, without the complex plane), including the Riemann Zeta function and an entire class of similar functions have zero's only on that line. In addition, it would be useful to show that all the zero's of any of the Generalized Riemann Zeta Functions are on the real plane, not the complex/imaginary plane. If these 2 things were proven, a lot of problems in Number theory would go away, and it would prove many problems that have been shown to be true if the Generalized Riemann Hypothesis is.

To restate this again, in English, (to the best of my ability, as some things cannot really be simplified,) The Generalized Riemann hypothesis says that all zeros (ie. Solutions) of a certain type of function, which includes the specific function ζ(s) (the Riemann Zeta Funtion,) are all on the line at which σ = 1/2. While this sounds incredibly esoteric, it has applications in Cryptography, and is so important (to number theorists) that it is sometimes called the holy grail of mathematics, and has a $1 Million Prize.