A "rectifiable $n$-varifold" is an equivalent class of $H^n$-measurable $n$-rectifiable sets. That is, the objects discussed in the last theorem. I will state the Allard regularity theorem, which says that under some conditions, a varifold which is nearly minimal is always a $C^{1,\alpha}$ submanifold. And then I will try to talk as much as possible about its proof.

References:

1. L. Simon, Lecture on Geometric Measure Theory.
2. R. Hardt & L. Simon, Seminar on Geometric Measure Theory.