I am no Physicist, but I enjoy reading about Physics. However reading about leading theories such as M-Theory and others they speculate about the existence of the Graviton.
In my past reading of General Relativity it was explained that Gravity was a result of the bending of space/time, so from that I have always thought of Gravity as a result not a force. If it is a result and not a force why would anyone be looking for a force carrying particle?
Where am I going wrong here?
The other forces are also just the result of "spacetime bending", just in a different way. There is no fundamental difference in the description of the other forces through gauge theories and gravity through relativity.1 The reason why it is often said that it is different is that our usual methods of quantizing a theory fail when applied to gravity. But to say "gravity is a result, not a force" is as right/wrong as saying "electromagnetism is a result, not a force". Polemically speaking, the notion of force or not force is the wrong idea to use when thinking about field theories and their quantization.
The currently known fundamental forces except for gravity can all be described in a framework that is analogous to that of relativity. It's all about modifying the ordinary derivative in a way that spits out something that transforms properly under the symmetry group of the theory, and is called gauge theory
General relativity has the diffeomorphism group of spacetime as this symmmetry, and uses the Levi-Civita connection to gain the covariant derivative. In this process, we gain a new dynamical object in the theory, namely the connection itself, or rather, the metric from which it is derived.
The strong force has the Lie group $\mathrm{SU}(3)$, the weak force has $\mathrm{SU}(2)$, and electromagnetism has $\mathrm{U}(1)$ as gauge groups, and they demand an Ehresmann connection to provide a gauge covariant derivative. To this connection, there is an associated connection form, usually called gauge field, which is a new dynamical object in the theory. The most commonly known form of this is the four-potential of EM.
The idea is always the same: There's some kind of transformation on our theory which should not change the physical laws. To make our ordinary derivative play nice with this theory, we introduce new objects added to it to kill the unwanted terms that are incurred when we would let the normal derivative as. These objects are the gauge fields. It's geometry just like in relativity, since this induces the notion of principal bundles.
Now, quantizing such theories, we find that there are particles associated to the gauge fields, and these are called force carriers, since the classical forces may be obtain from the tree-level Feynman graphs in which they are "exchanged" between two particles charged under the force. Since relativity also looks like such a theory (only having the rank 2 stress-energy tensor and not some rank 1 four-current as its "source"), it is natural to expect that quantizing the dynamical object of relativity will yield a graviton. Conversely, any field coupling to the stress-energy tensor is expected to yield a force indistinguishable from gravity.
1Well, there is in the sense that the metric as the dynamical object of relativity impacts the world a bit more...directly than gauge fields, but the point to make here is that the frameworks used to describe the forces are not really different.
That's a very nice answer by ACuriousMind. I would like to add something, though. GR is actually not like other gauge theories in some of its aspects (apart from having lots of similarities). For starters, it is background-independent and highly non-linear. In ordinary QFT we usually deal with perturbative expansions, which make sence only for weak-coupled renormalizable theories. Gravitons are usually obtained in linearized quantum gravity, which is just an approximation of GR (the metric is being expanded over some fixed background). Even this approximation is perturbatively non-renormalizable (the technical argument against GR quantization). But since actual GR is background-independent, there is no good way of introducing the perturbative expansion in the first place, because there is no true 'free gravity field' around which we would expand the interacting field perturbatively (another, conceptual argument). It makes some people doubt that the usual concept of 'gravitons' is applicable.
Force is a classical concept that is useful in modeling the mesoscopic world, i.e the world of classical thermodynamics, mechanics and electrodynamics.
Exchanged particles are quantum mechanical concepts which mainly work in small atomic size dimensions.
There is continuity in physics going from mesoscopic to the microscopic frameworks, and continuity in the physics theoretical models. Thus the very useful Feynman diagrams that show the virtual exchanges of mediating photons ( and bosons in general) can be seen as building up the mesoscopic forces we observe in our everyday dimensions.
If we forget General Relativity for the moment, and decide to quantize gravity, the way we quantized the electromagnetic field, the graviton will be the analogue of the photon in the microscopic framework which will at the mesoscopic limit appear as the gravitational force.
General relativity is applicable in a very much larger classical framework , call it macroscopic, but still the continuity in the theoretical models exists, at the mathematical limit of flat spaces and mesoscopic dimensions General Relativity gravity turns into Newtonian gravity, even though the concept of force does not exist in General Relativity, as you state.
When theorists quantize General Relativity it is natural to keep the terminology of "graviton" as a mediating particle, because quantization is expressed in Feynman diagrams and there exist mediating quanta, and it ensures the continuity with the flat space that is the realm of classical mechanics and classical electrodynamics.
In addition, there is an ongoing search for a theory of everything, where all four classical forces are quantized and unified at very high energies. In the beginning of the Big Bang all the exchanged bosons play the same interchangeable role, including the graviton.
