Closed surface intuition 
In topology, closed surface is simply defined to be the surface that has no boundary as opposed to open surfaces.

This is the layman's definition of closed surface. Example is notably a sphere.
But, I am unable to conceive the sense of the definition. How can a surface has no boundary? What does this actually mean? I'm not getting the intuition. Plz help. 
 A: If you had a spherical piece of paper, any point on the paper would be surrounded by paper on two dimensions. You could cut out a little circle with that point in the center. If you had a normal sheet of paper, most of the paper would be like that, but there'd be a boundary where the points only have paper on one side and you could only cut out a semicircle. That's what "boundary" means when dealing with surfaces.
Unfortunately, the definition you're showing is incomplete. A closed surface must also be compact. My favorite definition would be really difficult to explain, but if you're not using some really weird way of measuring distance, a simpler one will suffice. It must be closed and bounded (no relation to the "closed" and "boundary" I already mentioned). "Closed" here means that any point not on the paper is completely surrounded by points not on the paper, so you can't just have a normal sheet of paper where only the edge is missing so it technically has no boundary. "Bounded" means that it doesn't go on forever in any direction, so a plane wouldn't count.
Edit:
I think it's probably good to explain why compact is a thing. If you look at an open interval from zero to one, it's bounded. It doesn't go on forever. But you can take a continuous function of it (which preserves all the sorts of structures mathematicians love) and get something that goes on forever. For example, $f(x) = 1/x$ is continuous on that interval, and maps it to the open interval $(1,\infty)$. If you use a closed interval, you can't do that. Any continuous function of $[0,1]$ will map it to a bounded set. You could say $1/0 = \infty$, and topologists frequently do that, but adding an infinity like that messes around with the structure of the real line so much that you're less making $[0,1]$ infinite than you are making the real line finite.
Compact means that you're dealing with a set in which being finite is inherent to the structure in a way that can't be changed by something as simple as a continuous function.
A closed surface is one that doesn't go on forever but also doesn't have edges. It just loops around on itself like a sphere.
