The problem with stem cell research is pretty much the same question as asking whether or not abortion is correct.. unless they're asking about the inherent benefits of a particular line of stem cell research.
The problem lies, as Chrisobean said, with the way stem cells are harvested. Now, I have no problem if it comes from aborted embryos. I am neither condoning or flipping out over abortions, but if you're going to abort it anyhow, at least it should go to a good cause, a like a medical school cadaver.
Though if you have problems with stem cell research, I suggest you look up : Fetal Brain Grafts. That would probably drive some activists to violent action if it was actively performed here in the states. You may also want to bring up the point that if we're not doing stem cell research, that puts us at a very huge disadvantage to other countries such as China, and even Mexico.
As for the actual research itself. The OP was correct that it deals with undifferentiated cells, though I think more specifically with cells that have the ability to form colonies and continuously divide as opposed to limited progenitor cells (though the latter would still be a viable source of stem cell investigation).
It's not just for organs, though you hear all the big things are associated with organ repair, such as that one surgeon who shoved a bunch of stem cells into someone's heart in the hope that it would make the patients heart repair itself. There is a lot of interest in neural applications. Especially after finding out the brain does a limited amount of neurogenesis itself, and the spinal cord as well. It's a big topic.