A newtype is simply put an advanced lifeform, similar to the coordinators of Seed.
However they are NOT the same thing.
Where coordinators are "better" at everything than the naturals, newtypes just have a few abilities that regular humans lack.
The 2 most common, and important, of these:

1. The ability to sense other newtypes (this is what Mwu/Rau does in Seed and Rey/Neo in Destiny)
2. The ability to steer "funnels" (little laser shooting mini cannons that can be steered separately from the ship/gundam they are attached to) this is used by Mwu & Neo in their respective ships and by Rau in his Providence gundam at the end of Seed

Another thing that some newtypes are the victims of is mental disorders.
They become mentally instable and sometimes just plain crazy. The best example of this is probably Four Murasame from Z Gundam.
Again, this is not the case for all newtypes. Though I'd guess Rau is a bit affected by it...

What is the role of newtypes in Seed/Destiny?
It really isn't that big, it's more like a novelty IMO. Only the secondary characters are newtypes here and it doesn't affect the storyline.
In the older gundam series, newtypes were very important to the plot as all the main characters of the first 3 TV series (and the greatest enemies too for that matter) were just that.

So to finish it up, they are humans with special abilities who are only in Seed/Destiny for the sake of novelty and for older gundam fans to recognize.