Why are there only finitely many cubic fields of a given discriminant? Is this true for higher dimension too? 


What other invariants are needed to classify cubic fields? number of real and complex embeddings, Galois closure?...



Cubic Fields Up to Isomorphism
Your first two questions have already received good answers, so let me offer a few references about "classifying cubic fields", and point to some of my colleagues' interesting work in the subject. 


What information determines the field? This question turns out to be quite subtle. In general, even the Dedekind zeta function is not enough -- see [this](http://www.math.leidenuniv.nl/~desmit/prep/ants5.ps) paper of Bosma and de Smit (which does not address cubic fields though).


Scott Carnahan pointed you to the Davenport-Heilbronn theorem, which gives an asymptotic for the counting function of cubic fields with bounded discriminant. [This paper](http://arxiv.org/abs/1005.0672) of Bhargava, Shankar, and Tsimerman provides, among much else, a beautifully written self-contained introduction to the proof of this theorem.


For classifying cubic fields, knowing the discriminant is *not* enough. In particular, cubic fields with discriminant $D$ are, by class field theory, in bijection with subgroups of the class group of $\mathbb{Q}(\sqrt{D})$ of index 3, and it is expected (google the "Cohen-Lenstra heuristics") that these 3-torsion groups can be arbitrarily large.


Along these lines, an interesting paper was written by Guillermo Mantilla-Soler, see [here](http://arxiv.org/abs/1104.4598), where he studies the extent to which cubic fields are determined by their integral trace forms.


You can also study cubic fields by means of their quadratic resolvent, for example in [this paper](http://arxiv.org/abs/1003.1869) of Cohen and Morra. You can also look at the related 3-torsion problem in terms of Heegner points -- see [this](http://arxiv.org/abs/1005.1458) paper of Bob Hough, and there is forthcoming work of [Yongqiang Zhao](http://www.math.wisc.edu/~yqzhao/), who develops a powerful approach to counting cubic *function* fields in the language of algebraic geometry. Or, you can use [zeta functions](http://arxiv.org/abs/1102.2914).


There are a lot of ways to understand cubic fields -- and yet plenty of open questions remain.


