How do curves consist of points? According to Euclid, a point is something which has no dimensions. And we know that all curves of any type consists of points. Now this thing bothers me because if a point has no dimensions, i.e. in other words there is nothing, then how is it possible to draw any curve? 
The thing I could imagine is that maybe a point is not as Euclid thought. I mean a point can be thought of as a small line segment whose length is approaching zero but never becomes exactly zero. In this way, we can say that curves consist of points (the line segment with length approaching zero). But then it breaks the fact as given by Euclid that a point has no dimensions.  Please help me to get out of this dilemma.

You are confusing reality and the idealized world of maths. In an ideal world, a point has no dimensions. You could never actually draw a point. In reality, you can. You take a pencil, make a point on a piece of paper and say "that is the point". But the fact that you can see the point already means that is has some dimensions (the ink occupies a non-empty area in our real, three dimensional world). You have not actually drawn the point, but a representation of the (idealized) point.
Similar, if you draw a curve, you need to draw it with some thickness. This means you don't draw the actual idealized curve, but a real life representation of that curve.
Maths is founded in the real world, but goes one step further. Maths discovers the rules behind things, and abstracts them into an idealized world. This ideal world does not exist in reality, but it is nevertheless useful.
For example, infinity is a purely mathematical concept, yet it is tremendously useful for real-life applications, e.g. physics.
