Wolfram Alpha states,
The Skewes number (or first Skewes number) is the number Sk_1 above which pi(n) is less than li(n) must fail (assuming that the Riemann hypothesis is true), where pi(n) is the prime counting function and li(n) is the logarithmic integral
Why must the Riemann hypothesis be true?
Also, why does it change to 10^(10^(10^(10^3))) (Second Skewes number) when assuming that the Riemann hypothesis is not true?
If we do not rely on the truthness of the Riemann hypothesis (I am not sure, whether the general Riemann hypothesis would have to be true, which would be an even stronger requirement), we need a higher upper bound to guarantee a counter-example.
The second number is this upper bound. Regardless, whether the (genralized?) Riemann hypothesis is true, there must be a counterexmample not exceeding Skewes's second number.