If dark matter is a new type of particle, what does that imply? My understanding is that dark matter cannot be (or is at least highly unlikely to be) an exotic form of any known particle. On the other hand, articles about particle accelerators seem to say that the Higgs is the last piece missing in the Standard Model jigsaw puzzle. 
If dark matter is determined to be some form of new particle, what are the certain implications? Might such a discovery "stand to the side" of the Standard Model or would it certainly change the foundations? 
(Forgive my extremely lay understanding and vocabulary -- please feel free to correct my mistakes.)
 A: The conventional wisdom about dark matter is that it is likely to be a new kind of particle that is not part of the standard model. Basically, the reason for this is that most of the stable standard model particles interact electromagnetically (and so wouldn't be "dark"). 
The exception is neutrinos, and for a long time neutrino dark matter was considered to be a viable possibility, but it doesn't seem to work for several reasons. The main one is that neutrino dark matter would be "hot" (meaning that the particles would have had relativistic speeds in the not-too-distant past), whereas the way dark matter is observed to cluster gravitationally only seems to work if the dark matter is "cold."
For quite a while, people tried to come up with models in which the dark matter was ordinary matter (made of atoms), but that also doesn't seem to work for several reasons. In a Universe with a high enough density of atomic matter, the abundances of light elements produced in the early Universe would be quite different from what is seen. There's no good way to make this sort of matter "dark enough": even if you make it cold and neutral,\ it still interacts with radiation too strongly to remain hidden. Also, a Universe made of only atomic matter predicts a spectrum of fluctuations in the microwave background, and a bunch of other cosmological observables, that differ by a huge amount from what we observe.
So the leading theory is that the dark matter is a different sort of stable, neutral, weakly-interacting particle. Such a particle would necessarily be "beyond the standard model." 
Probably the least exotic possibility is that it's a supersymmetric particle. If supersymmetry is right, there are a lot of new particles out there waiting to be discovered. Most of them are unstable, but the lightest one is stable and would make an excellent dark matter candidate.
If supersymmetry is right, there's a good chance the LHC will detect it, so we may actually know the answer to this in the not too distant future. Then again, we may not.
A: Let me address this part of the question, as the physics part is covered  by Ted Bunn.

If dark matter is determined to be some form of new particle, what are the certain implications? Might such a discovery "stand to the side" of the Standard Model or would it certainly change the foundations? 

Up to now, progress in understanding particle physics has been happening not by completely discarding previous theories, but by assimilating them. The reason is that previous theories, as the standard model will become too in the future, were based on solid data. Being based on data means that they are just a shorthand of describing them. The old data still exists when new data appear, as the example of dark matter , so what will happen is that the standard model will be incorporated in any new theory describing matter in the microcosm.
Already string theories include the standard model within their structure so no "stand aside" will happen if they are found sufficient to describe all new data. String theories have a plethora of matter forms that could well explain/describe dark matter.
