Why does gravity seem to have two natures (force or warping of space and time)? In classical mechanics, gravity is regarded as a force but in general relativity it's a warping of space and time in presence of mass. Are these two definitions the same? Or is this a duality nature of gravity the same way we have duality of light being a particle and wave?
 A: 
In classical mechanics, gravity is regarded as a force but in general relativity it's a warping of space and time in presence of mass. Are these two definitions the same?

In pre-Einsteinian gravity, without other forces$$\frac{d^2}{dt^2}r=-\frac{\partial\Phi}{\partial r}$$describes acceleration due to gravity centred at the origin, with $r$ the polar radius, $t$ time and $\Phi$ the gravitational potential. The equivalent in general relativity is$$\frac{d^2}{d\tau^2}x^\mu=-\Gamma^\mu{}_{\nu\rho}\frac{dx^\nu}{d\tau}\frac{dx^\rho}{d\tau},$$with $\tau$ proper time and $\Gamma^\mu{}_{\nu\rho}$ the Christoffel symbols, which are one way to describe spacetime curvature. (For an alternative using the Riemann tensor, see here.) To relate these equations, take$$x^\mu=r,\,\frac{dx^\nu}{d\tau}\approx\delta^\nu_0\implies\frac{d^2}{d\tau^2}r\approx-\Gamma^r_{tt}.$$It can be shown $\Gamma^r_{tt}\approx\frac{\partial\Phi}{\partial r}$, recovering the force characterization of gravity.

Or is this a duality nature of gravity the same way we have duality of light being a particle and wave?

No, these are unrelated.
A: The two effects are one and the same.
Or more accurately, our experience of gravity as a force can be explained by gravity's effect of curving space.
It all hinges on Newton's first law of motion:

a body at rest will remain at rest and a body moving with constant velocity will continue moving with that velocity in a straight line until acted upon by an external force

So given something in space (a rock, a planet, a rocket etc.) that thing will continue moving in a straight line.
Since gravity affects space what is a "straight" line in the presence of gravity may not appear straight to us. Take Earth for example: the equator is a straight line that divides the planet into two hemispheres. However, if you zoom out you will see that the equator is a circle: a straight line on a sphere is a circle.
Similarly, a straight line in the presence of gravity bends towards the mass that causes the gravity. The object is merely following Newton's first law - since it has no engines nor are there any external force acting upon it it will continue moving in a "straight" line. From our point of view this straight line looks like it's bending towards the mass (either orbiting or falling onto the mass) but the object is not actually curving it's path, it is the space that the object is in that is curved. The object is actually still moving in a straight line inside a warped/curved space.
Similarly, the object is not actually accelerating towards the mass. It is moving with constant velocity in a "straight" line. But since gravity warps the space around the mass (the space near the surface is more compressed/dense), from our point of view the object appears to be accelerating.
It is because of gravity's effect on space that we experience it as a force that attracts objects.
However, as others have mentioned, this is just a model: a theory. Both explanations are just models. It just so happens that the warping of space theory is able to explain the force we perceive.
A: The equivalence principle states, that the inertial mass (which measures the resistance against acceleration, hence the inertia) and its gravitational mass (which measures the interaction with other bodies due to gravity) are equivalent. Floating weightless in a room, you would not be able to tell if it was drifting through space or falling towards the Earth or the sun. The equivalence principle guarantees the existence of a free falling coordinate system in which gravity vanishes. Gravity can therefore be interpreted as a force of inertia. The effect of gravity purely arises from the transformation to a different coordinate system, for example the room now resting on the surface of the Earth.
The two definitions are not the same. You could also describe electromagnetism as the curvature of spacetime, but it wouldn't make much sense. The reason is the different nature of both interactions. When Einstein discovered the equivalence of energy and mass, a simple contradiction to Newtonian gravity arose: The mass of planets and stars contributed to their gravity, but their movement and rotation didn't. But we can use an analogy in electrodynamics, where resting charges cause an electric field and moving charges cause a magnetic field. Special relativity already showed that they transform into each other under a Lorentz transformation. It seems like Einstein only had to add a "second gravitational field" to solve the contradiction and propose equations similar to the Maxwell equations for a revisited theory of gravity. This analogy still exists as a limit in General Relativity and is called "Gravitoelectromagnetism". The Lense-Thirring precession for example is the analogy of the Lorentz force.
But like electromagnetic waves, we would then get wave solutions for both our gravitational fields. While electromagnetic waves don't carry charge, those gravitational waves carry energy and according to the equivalence that caused our contradiction, this corresponds to a mass which generated gravity. It sounds strange, but gravity is a source of gravity. To describe this is what occupied Einstein for a decade and is one of the fundamental differences to forces like the electromagnetic interaction. Since then, gravity is no longer considered a force.
A: The classical and GR explanations of gravity are both models that describe the effect of gravity, one of which does so more accurately than the other. The two are not incompatible with each other in that sense- we simply have a single behaviour of matter (namely mutual attraction) that is modelled in two different ways.
That is quite different from quantum mechanics, where we have a single theoretical model of two distinct patterns of behaviour.
