I can only give you a brief explanation here. According to Newton, gravity was an invisible force that acted INSTANTANEOUSLY across space. He studied Copericus, and Kepler, and was able to come up with this equation
F = G M m / r^2
F = Force of gravity (N); G = universal gravitational constant (Nm^2/kg^2); M and m = masses (kg)
r = distance between the centers of mass (m)
Fast forward 300 years or so, and along comes Einstein. According to him, gravity was a change in the geometry of space, caused by the pressence of mass. He said that mass, warps or bends space [and time]. When a planet orbits the sun, it is simply following the curve in the space around the sun, caused by the sun's mass.
As to which one is correct, well, that depends on how accurate you want to be. When NASA send out space probes to the planets, they use Newton's law, which is perfectly adaquate. But, Newton's law has a problem. This was highlighted by the orbit of Mercury, where the effect is most prominent. Newton's equation predicts that the planets should move in elipses. To move in an elipse, the planet would have to start AND END its orbit at the SAME PLACE. But, no planet does that. The planet 'overshoots' a little bit after each orbit. This has the effect of causing the plane of the orbit to rotate, slowly around the sun. This movement is called precession. Newton was aware of the problem with Mercury, but put the effect down to observational errors. The movement is only about 43 secs of arc PER CENTURY! Its tiny.
Einstein's equation correctly allows for this movement. So Einstein's is just a more accurate way to do things. They are both correct within their range of use. However, Einstein showed that gravity CANNOT act instantaneously. Any change in the gravitational field from the sun can only travel AT THE SPEED OF LIGHT, and would thus take about 8 minutes to reach us. So if the sun were to suddenly disappear, everything would seem fine here on earth. Then, 8 mins later, we would suddenly be aware that the sun was no longer there.