What is the geometric interpretation of Euler characteristic? By this question, I know ,in the high dimensional $(\ge 3)$, the Euler characteristic is not $2-2g$, $g$ is genus. When dimension equal to $2$ , genus is the holes of surface. It is easy to understand, but when dimension $\ge3$, what is the geometric interpretation of Euler characteristic .
 A: If you want to understand the higher equivalents of the genus, you should rather take a look at Betti numbers. It is a defining motivation that the $n$th Betti number $b_n$ counts "$n$-dimensional holes" in some sense (see the article for more examples). 
For surfaces it turns outs the the first Betti number $b_1$ is always even, and we can define the genus of a surface by $b_1=2g$. Since the genus is also often introduced as counting the number of "holes" in a surface, this shows that this notion of "hole" should be treated with caution. In the genus version, we consider that a torus has one hole, which is visually intuitive. In the Betti number version, we say it has two holes, one coming from the tubular shape, and one coming from the circular hole in the middle. Of course both $b_1$ and $g$ have rigorous definitions, this business of "hole-counting" is just supposed to provide some intuition.
Once we have all the Betti number $b_n$, the Euler characteristic is 
$$\chi = b_0 - b_1 + b_2 - \dots + (-1)^n  b_n$$
where $n$ is the dimension of your manifold (in general, the Euler characteristic is defined as long as the Betti numbers are all $0$ for indices large enough).
In the case of (connected) surfaces, $b_0=b_2=1$ for connectedness/duality reasons, so $\chi = 1 -2g + 1=2-2g$, and the Euler characteristic only depends on the genus. But in general it will be more complicated.
The Euler characteristic has a lot of important and magical properties (such as multiplicativity:  $\chi(X\times Y)=\chi(X)\chi(Y)$), but I find that it is much less intuitive to understand at first than Betti numbers. The fact that it is introduced quite early in a lot of geometry courses is possibly a historical artifact, since the Euler characteristic was defined way before the Betti numbers.
For other geometric interpretations of the Euler characteristic, you can just take a look at the wikipedia article, which for instance mentions links with homological invariants of vector bundles (Euler classes), or the generalized Gauss-Bonnet theorem.
A: There are many geometric definitions of Euler characteristic, here are 2 examples:
1) Suppose $M$ is a manifold. Take the diagonal $\Delta: M \to M\times M$, and consider another embedding of $M$ into $M\times M$. Then in most cases these 2 embeddings will intersect transversely and counting points of intersection with appropriate orientation gives exactly $\chi(M)$.
2) This one is much more popular equivalent definition:Let $M$ be a compact smooth manifold. Let $V$ be a vector field on $M$ with isolated zeroes. Then:
$\chi(M)$ is the analytical index of the vector field $V$.
