Motivated by Tukey classification problems and building on work in Part 1, we develop a new hierarchy of topological Ramsey spaces
R
α
\mathcal {R}_{\alpha }
,
α
>
ω
1
\alpha >\omega _1
. These spaces form a natural hierarchy of complexity,
R
0
\mathcal {R}_0
being the Ellentuck space, and for each
α
>
ω
1
\alpha >\omega _1
,
R
α
+
1
\mathcal {R}_{\alpha +1}
coming immediately after
R
α
\mathcal {R}_{\alpha }
in complexity. Associated with each
R
α
\mathcal {R}_{\alpha }
is an ultrafilter
U
α
\mathcal {U}_{\alpha }
, which is Ramsey for
R
α
\mathcal {R}_{\alpha }
, and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on
R
α
\mathcal {R}_{\alpha }
,
2
≤
α
>
ω
1
2\le \alpha >\omega _1
. These form a hierarchy of extensions of the Pudlak-Rödl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our Ramsey-classification theorems to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to
U
α
\mathcal {U}_{\alpha }
, for each
2
≤
α
>
ω
1
2\le \alpha >\omega _1
: Every nonprincipal ultrafilter which is Tukey reducible to
U
α
\mathcal {U}_{\alpha }
is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of rapid p-points. Moreover, we show that the Tukey types of nonprincipal ultrafilters Tukey reducible to
U
α
\mathcal {U}_{\alpha }
form a descending chain of rapid p-points of order type
α
+
1
\alpha +1
.