As an object travels in an elliptical orbit around the sun, there is a point in the orbit where the object is closest to the sun (obviously). This point is called the perihelion of the orbit. The precession of perihelion is an effect whereby the perihelion actually occurs at a different angle, from one orbit to the next.
Contrary to some reports, general relativity is not the only contributing factor to the precession of perihelion. The gravitational pull of other bodies in the solar system, and the slight bulge of the sun about its equator, both contribute to the effect - even if only considering Newton's laws. The number associated with Mercury's orbit, 43 arcseconds per century, is actually the difference between the Newton model and the observed values, and this difference is what is accounted for by general relativity.