Genus of a topological surface I am not understanding the geometry of the actual definition of Topological genus of a surface.
In Wikipedia I have found the following definition:

The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected.It is equal to the number of handles on it.

I am unable to understand the geometry. Can someone please describe this?
 A: This is probably best done with a sequence of examples.
On a sphere, any closed curve (read: loop) we cut along will disconnect the sphere:

I've shown this for one particular loop, but it's not hard to convince yourself it's true of any loop. So we can cut along $0$ loops without disconnected the sphere, making it genus $0$.
Next consider a torus. It's easy to find a loop which we can cut along so that the resulting surface is still connected, see here:

Of course, if we try to cut along any second loop, we will necessarily be left with a shape in two pieces.
Since we can cut along $1$ loop without disconnecting the shape, we say it has genus $1$.
Finally, let's consider a genus $2$ surface:

Again, you should convince yourself that any three cuts will result in multiple pieces, while here we were able to make $2$ cuts without disconnecting the surface. So this surface has genus $2$.
The case of higher genera (the plural of "genus") is similar.

Edit: Oh, and what does this have to do with handles? It turns out every (connected, oriented, compact) surface can be deformed into a sphere with handles. See this photo of a genus $3$ surface from wikipedia:

Notice we can make a cut in the middle of every handle while leaving the figure connected, but we can't cut anywhere else. This is some informal proof that the genus is equal to the number of handles your surface has.

I hope this helps ^_^
A: The genus of a connected, orientable surface is intuitively the number of "holes" in that surface. For instance: the sphere has genus 0, the standard torus has genus 1, etc. The Wikipedia page on Genus g surfaces has some nice visualisations.
In terms of the definition you've given, it's useful to consider that a "cutting" (under the conditions specified) would reduce a genus-n surface to a genus-(n-1) surface, either by cutting between two holes and joining them into one, or by cutting between a hole and the outside of the surface, eliminating that hole. Notice that any cutting on the sphere (the genus 0 surface) splits it into two connected components, rendering it disconnected.
We can then see that, for a genus g surface, it will take g cuttings to reduce it to the sphere, at which point any further cuttings will render it disconnected. This is effectively what your definition is trying to elucidate.
