Centripetal effect or curved space time I think I understand both the centrepedal effect and Einsteins curved space time. However I am confused about which best describes the motion of a planet ( or other orbiting body ). Simply put, does the earth experience any centrepedal effect or does it just follow the geodesic line and therefore not experience any centrepedal forces. Can someone please explain or let me know that science has yet to answer this question.
 A: The simple answer is yes.
If the earth did not experience centrepedal force it would fall into the sun.
If the earth did not experience curved space it would move away from the sun and not remain in orbit.
EDIT:  Think of these 2 forced creating an exact balance, in the case of your example.  Other cases are not exact balances:  Mars' moons Phobos and Deimos are good examples.  Phobos does not have enough centrepedal force to keep it in orbit and will eventually fall to the surface of Mars.  Deimos has too much centrapedal force and will eventually escapr Mars' gravitational field.
A: You can explain the motion of the Earth around the Sun either by a centipetal gravitational force or by spacetime curvature. It isn't the case that one description is wrong and one is right, but rather that they are both mathematical models that describe the motion in different ways.
To illustrate this suppose we replace the Sun by a positive charge and the Earth by a negative charge. A negative charge will orbit a positive charge in the same way that a planet orbits the Sun. Indeed this principle was the basis of the Bohr model for the hydrogen atom. In a model like this we would be quite happy to consider the orbital motion to be due to a centripetal force (the electromagnetic force), though as it happens you can describe the EM force using the concepts of curvature.
The attraction of general relativity is that it explains exactly what the gravitational force is and how it arises. For example Newton's law of gravitation tells us that the centripetal force will be:
$$ F_\text{Newton} = \frac{GMm}{r^2} $$
This is a nice simple equation, and indeed we would expect an inverse square force like this to apply so it's a reasonable approach to studying orbital motion. However when we use general relativity to look more deeply we find that the force is actually:
$$ F_\text{GR} = \frac{GMm}{r^2\sqrt{1-\frac{2GM}{c^2r}}} $$
We could continue to work with a centripetal force $F_\text{GR}$ but the equation has become a bit messy and in any case it only applies in the specific case of spherical symmetry. If we decide to abandon the idea of a force and use spacetime curvature instead we find describing the motion becomes conceptually simpler (though the details of the maths can be hard!).
The point is that if you're a physicist then what you're trying to do is predict how objects will move in a gravitational field. In some circumstances it's easier to use a centripetal force and in other circumstances it's easier to use a curvature approach. Whatever approach you choose, as long as it ends up giving the correct answers it isn't wrong.
