As far as my understanding goes the answer is no, and I will try to explain why and clarify the list of comments (I have little reputation so I cannot comment there) and give you a partial answer. I hope I do not patronise you, since you may now already part of it.
First of all, as Torsten said, it depends what you understand for classification. In this context a torus $T$ of dimension $r$ is always an algebraic variety isomorphic to $(\mathbb{C}^*)^r$ as a group. A complex algebraic variety $X$ of finite type is toric if there exists an embedding $\iota: (\mathbb{C}^\ast)^r \hookrightarrow X$, such that the image of $\iota$ is an open set whose Zariski closure is $X$ itself and the usual multiplication in $T=\iota((\mathbb{C}^\ast)^r)$ extends to $X$ (i.e. $T$ acts on $X$).
Think about all toric varieties. It is hard to find a complete classification, i.e. being able to give the coordinates ring for each affine patch and the morphisms among them for all toric varieties.
However, when the toric varieties we consider are normal there is a structure called the fan $\Sigma$ made out of cones. All cones live in $N_\mathbb{R}\cong N\otimes \mathbb{R}$ where $N\cong \mathbb{Z}$ is a lattice. A cone is generated by several vectors of the lattices (like a high school cone, really) and a fan is a union of cones which mainly have to satisfy that they do not overlap unless the overlap is a face of the cone (another cone of smaller dimension). There is a concept of morphism of fans and hence we can speak of fans 'up to isomorphism' (elements of $\mathbf{SL}(n,\mathbb{Z})$). Given a lattice N, there is an associated torus $T_N=N\otimes (\mathbb{C}^*)$, isomorphic to the standard torus.
Then we have a 1:1 correspondence between separated normal toric varieties $X$ (which contain the torus $T_N$ as a subset) up to isomorphism and fans in $N_\mathbb{R}$ up to isomorphism. There are algorithms to compute the fan from the variety and the variety from the fan and they are not difficult at all. You can easily learn them in chapter seven of the Mirror Symmetry book, available for free. Given any toric variety (even non-normal ones) we can compute its fan, but computing back the variety of this fan many not give us the original variety unless the original is normal. You can check this easily by computing the fan of a $\mathbf{V}(x^2-y^3)$ (torus embedding $(t^3,t^2)$) which is the same as $\mathbb{C}^1$ but obviously they are not isomorphic (the former has a singularity at (0,0)). In fact, since there are only two non-isomorphic fans of dimension 1 (the one generated by $1\in \mathbb{Z}$ and the one generated by 1 and -1) we see that there are only three normal toric varieties of dimension 1, the projective line and the affine line, and the standard torus.
The proof of this statement is not easy and to be honest I have never seen it written down complete (and I would appreciate any reference if someone saw it) but I know more or less the reason, as it is explained in the book about to be published by Cox, Little and Schenck (partly available) This theorem is part of my first year report which is due by the end of September, so if you want me to send you a copy when it is finished send me an e-mail.
So, yes, in the case of normal varieties there is some 'classification' via combinatorics, but in the case of non-normal I doubt there is (I never worked with them anyways).
Become a toric fan!.
This is a partial answer. Let $X$ be the given abstract variety. I think the question is equivalent to asking whether all maximal tori in the group $\mathrm{Aut}(X)$ are conjugate. When $X$ is complete, this is a linear algebraic group, so all maximal tori are conjugate, and the answer is affirmative. (See Cox's famous paper on the homogeneous coordinate ring.)
If $X$ is not complete, the automorphism group may of course be infinite-dimensional, but perhaps you can argue by compactifying.
Best Answer
David Cox has some nice expositions on toric varieties on his web page here. Cox is also one of the authors of the book "Toric Varieties", which is a very readable, yet comprehensive introduction to toric varieties. The first chapter here should provide you with enough motivation and examples for your talk. Then there is also chapter 1 in Fulton's book, which is the classic reference on the subject.
As for the motivational examples, you should look for examples that show the real power of toric varieties: That abstract algebro-geometric constructions can uaually be viewed very concretely by working with the defining combinatorial data (e.g. the fan). Some of these examples might do the trick:
1) The quadric surface $\{xy-zw=0\}$ in $\mathbb P^3$ and its affine cone in $\mathbb A^4$
2) The singular quadric $y^2=zw$ in $\mathbb A^3$.
3) Hirzebruch surfaces
4) Toric blow-ups and subdivisons of the fan
In the basic examples 1)-3), it is straightforward to write out the action of the torus, and see directly how monomials in the coordinate ring relates to the lattice points in the dual cones. Also, in the projective examples you can see how gluing the affine toric varieties works in terms of the fan data.
These examples demonstrate typical features of toric varieties, for example that their ideals are generated by binomials and their Chow ring is generated by the torus invariant subvarieties.
I like the Hirzebruch surface example because you can somehow 'see' the $\mathbb P^1$-bundle structure in the defining polytope and it is intuiticely clear that any toric surface is a blow-up of either $\mathbb P^2$ or a Hirzebruch surface. Moreover, I think it's pretty cool that you can view birational morphisms of toric varieties (e.g., resolution of singularities) as subdivisions of the defining fans. The example in this MO thread illustrates this. Another interesting example is the affine cone $Z(xy-zw=0)\subset \mathbb A^4$ which gives a nice combinatorial interpretation of the Atiyah flop.