Keep in mind that in The Fellowship Of The Ring, Gandalf tells Frodo,
On foot even Glorfindel and Aragorn together could not withstand all the Nine at once.
(Fellowship of the Ring, Book II, Chapter 1)
Αnd Glorfindel is considerably more powerful than Aragorn. Furthermore, Aragorn with the Rohirrim is very nearly defeated by the combined army of Orcs and Men at Helm's Deep; they are saved only by the timely return of Erkenbrand and the intervention of the Huorns.
You can read more about the war itself in Appendix A, in the section "The North-kingdom and the Dunedain".
As far as "The Witch-king and what army?" we're told in that section,
[Angmar] lay on both sides of the [Misty] Mountains, and there were gathered many evil men, and Orcs, and other fell creatures. The lord of that land was known as the Witch-king ... who came north with the purpose of destroying the Dunedain in Arnor, seeing hope in their disunion, while Gondor was strong.
So, I'd say, there was a mix of forces that were used to overwhelm Rhudaur, and Arthedain, and ultimately Arnor; and the Witch-king did this (as he did everything) to further Sauron's interest, although whether it was at Sauron's direct orders is not specified.