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?
[Physics] Why does gravity seem to have two natures (force or warping of space and time)
curvaturedualityforcesgeneral-relativitygravity
Related Solutions
Luboš's answer is of course perfectly correct. I'll try to give you some examples why the straightest line is physically motivated (besides being mathematically exceptional as an extremal curve).
Image a 2-sphere (a surface of a ball). If an ant lives there and he just walks straight, it should be obvious that he'll come back where he came from with his trajectory being a circle. Imagine a second ant and suppose he'll start to walk from the same point as the first ant and at the same speed but into a different direction. He'll also produce circle and the two circles will cross at two points (you can imagine those circles as meridians and the crossing points as a north resp. south poles).
Now, from the ants' perspective who aren't aware that they are living in a curved space, this will seem that there is a force between them because their distance will be changing in time non-linearly (think about those meridians again). This is one of the effects of the curved space-time on movement on the particles (these are actually tidal forces). You might imagine that if the surface wasn't a sphere but instead was curved differently, the straight lines would also look different. E.g. for a trampoline you'll get ellipses (well, almost, they do not close completely, leading e.g. to the precession of the perihelion of the Mercury).
So much for the explanation of how curved space-time (discussion above was just about space; if you introduce special relativity into the picture, you'll get also new effects of mixing of space and time as usual). But how does the space-time know it should be curved in the first place? Well, it's because it obeys Einstein's equations (why does it obey these equations is a separate question though). These equations describe precisely how matter affects space-time. They are of course compatible with Newtonian gravity in low-velocity, small-mass regime, so e.g. for a Sun you'll obtain that trampoline curvature and the planets (which will also produce little dents, catching moons, for example; but forget about those for a moment because they are not that important for the movement of the planet around the Sun) will follow straight lines, moving in ellipses (again, almost ellipses).
Gravity is viewed as a force because it is a force.
A force $F$ is something that makes objects of mass $m$ accelerate according to $F=ma$. The Moon or the ISS orbiting the Earth or a falling apple are accelerated by a particular force that is linked to the existence of the Earth and we have reserved the technical term "gravity" for it for 3+ centuries.
Einstein explained this gravitational force, $F=GMm/r^2$, as a consequence of the curved spacetime around the massive objects. But it's still true that:
Gravity is an interaction mediated by a field and the field also has an associated particle, exactly like the electromagnetic field.
The field that communicates gravity is the metric tensor field $g_{\mu\nu}(x,y,z,t)$. It also defines/disturbs the relationships for distances and geometry in the spacetime but this additional "pretty" interpretation doesn't matter. It's a field in the very same sense as the electric vector $\vec E(x,y,z,t)$ is a field. The metric tensor has a higher number of components but that's just a technical difference.
Much like the electromagnetic fields may support wave-like solutions, the electromagnetic waves, the metric tensor allows wave-like solutions, the gravitational waves. According to quantum theory, the energy carried by frequency $f$ waves isn't continuous. The energy of electromagnetic waves is carried in units, photons, of energy $E=hf$. The energy of gravitational waves is carried in the units, gravitons, that have energy $E=hf$. This relationship $E=hf$ is completely universal.
In fact, not only "beams" of waves may be interpreted in terms of these particles. Even static situations with a force in between may be explained by the action of these particles – photons and gravitons – but they must be virtual, not real, photons and gravitons. Again, the situations of electromagnetism and gravity are totally analogous.
You ask whether the spacetime is the force field. To some extent Yes, but it is more accurate to say that the spacetime geometry, the metric tensor, is the field.
Concerning your last question, indeed, one may describe the free motion of a probe in the gravitational field by saying that the probe follows the straightest possible trajectories. But where these straightest trajectories lead – and, for example, whether they are periodic in space (orbits) – depends on what the gravitational field (spacetime geometry) actually is. So instead of thinking about the trajectories as "straight lines" (which is not good as a universal attitude because the spacetime itself isn't "flat" i.e. made of mutually orthogonal straight uniform grids), it's more appropriate to think about the trajectories in a coordinate space and they're not straight in general. They're curved and the degree of curvature of these trajectories depends on the metric tensor – the spacetime geometry – the gravitational force field.
To summarize, gravity is a fundamental interaction just like the other three. The only differences between gravity and the other three forces are an additional "pretty" interpretation of the gravitational force field and some technicalities such as the higher spin of the messenger particle and non-renormalizability of the effective theory describing this particle.
Best Answer
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.