An addendum to the answers of Daniel Grumiller and sb1:
The major difference of the gravitational field and other fields is that according to general relativity the gravitational field defines space and time and therefore defines the relation of events. It is true that it is possible to do an "arbitrary" split of a certain linear approximation of the gravitational field into a "flat background" and "waves" propagating on this background. In principle this kind of reasoning is a violation of the very idea that the gravitational field defines the background of spacetime in a holistic way, and it was the subject of a lot of discussions if this approximation is of any use.
This is considered to be settled by the observational evidence that bistar systems loose energy in exact the way that the "graviational wave approximation" predicts, as cited by Daniel Grumiller.
The existence of gravitons is a conjecture based on the assumption that gravitational waves exhibit the same quantum nature as classical waves, e.g. waves in classical electromagnetism. At the basis of this conjecture is the idea that it should be possible to split the gravitational field in a part defining the background, and then having gravitational waves propagating on this background and exhibiting the same wave-particle duality as other waves. It would then be possible to treat quantum gravitational effects in a semi-classical approximation.
Since there is no observational evidence for this, this conjecture is still the subject of controversy.
Are these different communities of physicists?
Some have a strong believe in the existence of gravitons, some think that quantum gravity needs a bigger conceptual step than only gravitions, and some do both, so, yes, there are different communities believing different things.
Does relativity explain only part of the story of masses acting under gravity?
It explains everything in a classical setting with not too strong forces alright, but does not exlpain quantum effects or what happens when forces get so strong that singularities occur.
Is gravity a force or not? Is it only an apparent force or not?
It is a force, it is an apparent force in the sense that classical GR says that you feel it because you live in an accelerating reference frame. Both statements are valid in the classical setting and are independent of the quantum nature of gravity, and in particular of the existence of gravitons.
Can such an apparent force 'generate' exchange particles? Are the exchange particle and geometric models both different views of the same underlying truth?
Yes, see above (geometric model = classical setting, exchange particle = semi-classical approximation).
why can't the other forces be explained away similarly? Or is that what is happening with all this talk of small extra dimensions?
The gravitational field is fundemantally different from other fields (see above), and this has no connection to extra dimensions.
I'd appreciate any illumination on this matter, or suggested reading (preferably at the 'popular science' or undergraduate level).
The problem is that if you are able to ask this question, you're already beyond the popular science level. I'd really like to recommend to you an introducory class on QFT and one on GR, there you'd get the best answer to your question :-)