The one that Gil Kalai mentions is the best one.

Here is a couple of other applications, although they are not really applications of Dvoretsky's theorem per se.

Algebraic topology:
Milman (or Gromov? or Makeev?) realized that Dvoretsky's theorem follows from the (now known to be false) Knaster conjecture in algebraic topology, in fact the conjecture implies huge improvement on the bound on the dimension of the almost spherical sections of Dvoretsky's theorem. (The dependance on d is tight but the dependance on epsilon is pretty bad).
What ended up happening is that a counterexample to Knaster's conjecture comes from convex geometry. This is a result of Kashin and Szarek.
But there are weak versions of the Knaster conjecture that could be true. There are some recent related results by Karasev and Dolnikov and no so recent by Burago (the known to be false, so-called topological Dvoretsky's theorem).

There are a couple of Nonlinear versions of Dvoretsky's theorem, one by Bourgain, Figiel and Milman, and a recent one conjectured by Tao and proved by Naor and Mendel.

Computer Science:
The finite nonlinear Dvoretsky's theorem says that any finite space contains large subspaces that embed in $l_2$ with low distortion (when the distortion is a more than 2 or 3 one can actually find polynomial sized subspaces, while if one insists on $1+\epsilon$ distortion then the size is logarithmic). In "Ramsey partitions and proximity data structures", Naor and Mendel found some applications the partition trees of Calinescu, Karloff and Rabani tailored for this theorem to "proximity data structures" of Thorup and Zwick (CS).

Metric Geometry:
The compact Dvoretsky's theorem claims that any compact space contains subspaces of large Hausdorff dimension that embed with constant distortion in an ultrametric. (Actually all the nonlinear versions factor the map into Hilbert space through an ultrametric which embeds isometrically in Hilbert space, this idea is from Linial, Bartal, Naor and Mendel - I believe).
With this result, Naor observed that for any compact space of Hausdorff dimension more than n there is a surjective Lipschitz map to the n-dimensional unit ball of Euclidean space. (Why? This was easy to see).

I have often wondered about a Dvoretsky type theorem for Riemannian manifolds. Ten points for a good conjecture.

1more comment