According to Newton's laws of gravity, Kepler's laws are exact for the 2-body problem and the elliptic curves are exactly periodic. The perihelion – point on the orbit where the Mercury is closest to the Sun – is exactly at the same place.
But this is a very special feature of the $A/r$ potential. If you consider the potential to be $A/r+B/r^2$, you would find out that the perihelion doesn't stay at the same place. General relativity basically allows you to replace the gravitational field of the Sun by the $A/r+B/r^2$ in a certain approximation; the extra term arises because the gravitational field is "nonlinear" and acts on itself i.e. Einstein's equations of general relativity don't quite obey the superposition principle. The second term $B/r^2$ is particularly important if $r$ is small enough, it's very strong for the planets that are closest to the Sun. Mercury is ideal to see it.
When you calculate how much the perihelion of Mercury changes its location in a century, you will get something like 40 arc seconds per century, and it exactly agreed with some discrepancies between theory and experiments. Well, the total measured precession is 574 arc seconds per century, but most of it was explained by the action of other planets etc. already before Einstein. Remarkably enough, they were able to determine that 40 arc seconds per century were still wrong.