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Fano varieties are defined by the ampleness of $-K_X$, and a rough statement of a step in the Mori program is to check whether a variety is a Fano-fibered one. By that reason, Fano ones are important in classification problems.
From somewhere, I have read the phrase "Fano varieties have rich geometry". What is geometry in this context? How does their geometry help us to classify the Fano varieties?
Let $X$ be an smooth Fano variety of dimension $n$, defined over $\mathbb{C}$ to simplify.
Here is a non exhaustive list of some nice properties of Fano varieties, which of course can be complemented by the experts here in mathoverflow.
From Kodaira vanishing theorem, $\operatorname{H}^i(X,\mathcal{O}_X)=0$ for $i>0$. In particular, $\operatorname{Pic}(X)\cong \operatorname{H}^2(X,\mathbb{Z})$. In particular the Picard number of $X$ equals to the second Betti number and it is a topological invariant. Moreover, for every $n$ there only finitely many families of Fano manifolds.
The Minimal Model Program for a variety with negative Kodaira dimension is expected to end up with a fiber type morphism whose general fibers are (singular) Fano varieties. Moreover, the Cone Theorem easily implies that Fano manifolds has rational polyhedral Mori cone. Even better, they are Mori Dream Spaces.
A lot of examples can be explicitly constructed by using toric varieties. They are combinatorial and explicit computations can be done. Also, since general elements of the anticanonical complete linear system $|-K_X|$ are Calabi-Yau varieties, Fano varieties appear in Mirror Symmetry and explicit constructions can be done in the toric case by looking at polytopes.
Fano varieties are rationally connected, so there are plenty of rational curves on $X$, their first fundamental group is trivial, there are not non-trivial étale coverings, etc.
The singular analogs of these facts are very deep. Maybe you can look at the recent article of C. Birkar "Singularities of linear systems and boundedness of Fano varieties" for (1), BCHM article "Existence of minimal models for varieties of log general type" for (2), D. Cox and S. Katz wrote "Mirror Symmetry and Algebraic Geometry" which talks about (3) at some point, and Kollar's article "Which are the simplest algebraic varieties?" is related to (4).
Some lists to look for examples/classifications are the Graded Ring Database (for toric Fano 3-folds and toric Del Pezzo surfaces), the original works of Iskovskikh and Mori/Mukai (see also this article), and also there is a nice list of Fano 4-folds of Fano index $r>1$ here (see also the references therein). I recently assisted to a talk of T. Coates about a joint work with E. Kalashnikov where they found 138 new Fano 4-folds, he also explained how this fits into a program (joint work with Corti, Galkin, Golyshev, Kasprzyk, and others) to find and classify Fano manifolds using mirror symmetry.
