Obiwan, the "Standard model" of particle physics was developed mostly in the 1970s. It does explain how almost all of the different particles we know of interact and, as you say, "move" -- though most physicists would probably not use quite the term "move," since it implies more classical behavior than particles actually engage in. (In many cases, a particle acts like it's several places at once, etc., like a wave).
The thing is, when they were putting together the standard model, it didn't quite work out with just what they knew. The math did not fit together. And then, somebody had the idea that if there were this additional field/particle, the Higgs boson, the rest of the math would all work out. So they posited it, and everything was great, except there was this additional particle that had never been seen. As you can imagine, that was a substantial loose end. So today vindicates that bold hypothesis, and along with it, vindicates the standard model. (Well, almost. While they seem to have found the Higgs, one part of the data is not quite what was anticipated, and this might suggest an additional particle they didn't quite expect, but hey, that's exciting, right?)
Now, the Standard Model is, itself, not _quite_ complete. While it fully describes the interactions of almost all particles and "forces" -- the electromagnetic, the strong, the weak -- as others have said, it does not incorporate gravity. Gravity is expected to work, from a QFT point of view, through the graviton (as ulytau pointed out), but the graviton / gravitational field can't be incorporated into the Standard Model through any means yet discovered. Consequently, nobody can fit the gravitational force between particles into the picture.
As rokakoma points out, of course, there is some chance -- though I think most physicists would say a small one -- that gravitation isn't supposed to be folded into the theory in the expected way, and that they interact in some other, so-far unanticipated way. One thing for sure is that the geometric aspect of gravity / general relativity is one of the factors that substantially complicates working with it.