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!.
A few of the more obvious ones:
* Resolution of singularities in characteristic p
*Hodge conjecture
* Standard conjectures on algebraic cycles (though these are not so urgent since Deligne proved the Weil conjectures).
*Proving finite generation of the canonical ring for general type used to be open though I think it was recently solved; I'm not sure about the details.
For vector bundles, a longstanding open problem is the classification of vector bundles over projective spaces.
(Added later) A very old major problem is that of finding which moduli spaces of curves are unirational. It is classical that the moduli space is unirational for genus at most 10, and I think this has more recently been pushed to genus about 13. Mumford and Harris showed that it is of general type for genus at least 24. As far as I know most of the remaining cases are still open.
Best Answer
My favourite is Oda's Strong Factorization Conjecture:
Note that if you are allowed to intermingle the blow-ups and blow-downs (the weak version) it has been proved. In fact, it was proved for general varieties in characteristic 0 using the toric case:
Torification and Factorization of Birational Maps. Abramovich, Karu, Matsuki, Wlodarczyk.
A conjectural algorithm for computing toric strong factorizations can be found in the following arXiv article:
On Oda's Strong Factorization Conjecture. Da Silva, Karu.