For your first question, I take it that you are interested in orbit closures of nilpotent $n \times n$ matrices. I don't know anything about nilpotent orbits for other Lie algebras, but some stuff is in the references below.

As to your first question, it depends on what you want. Do you want an ideal of polynomials vanishing on these orbits? Or do you want the radical of this ideal? The first case can be done without too much work, but the second question is difficult (as far as I know).

We need a way to index these orbits. They are of course indexed by partitions like you say, given by the block sizes of the Jordan blocks. One way is to say that the orbit corresponding to the partition $\lambda$ is given by the conditions $\lbrace A \mid \ker A^i = \lambda_1 + \cdots + \lambda_i\rbrace$, in which case an ideal which set-theoretically defines the orbit closure is given by the equations

$\dim \ker A^i \ge \lambda_1 + \cdots + \lambda_i$

To get explicit equations, let A be a generic matrix of variables $x_{i,j}$. The inequality above is the same as saying that

$\operatorname{rank} A^i \le n - (\lambda_1 + \cdots + \lambda_i)$,

which we can write as polynomial equations by requiring that the $(N_i+1) \times (N_i+1)$ minors of $A^i$ are all 0, where $N_i = n-(\lambda_1 + \cdots + \lambda_i)$.

As for finding the radical of this ideal, one possible reference for this stuff is Chapter 8 of Jerzy Weyman's book *Cohomology of Vector Bundles and Syzygies*. Some of this is based on the material in a paper in which he calculates the radical of the ideal for certain nilpotent orbits (and proves some results about them in general): http://arxiv.org/abs/math/0006232 . What you're interested in should be in the paper, though it's heavy on sheaf cohomology (hopefully you like that). For algebraic properties of these coordinate rings like normality, Gorensteinness, rational singularities, see the book.

EDIT: I should have linked to the paper that David mentions since it contains more complete results than the one I included. However, the calculating of generators for the radical ideal (but not minimal ones) is done in Section 8.2 of the book, so I still recommend looking at it. In particular, all of the relevant techniques are treated from scratch in the earlier chapters. In particular, Chapter 5 is crucial.