How is it possible for a dot to have no dimension, while a line can have 1 dimension? In mathematics it is said that a "dot" has no dimension.
On a different context it is said that a line is made by joining different "dots".
Then how do line have only one dimension when its made up of something which has no dimension?? 
 A: The first thing you need for this question to make sense is a definition of dimension. One convenient and commonly used notion is the Hausdorff dimension. The definiton may be too technical and beyond the level you seem to be asking at, but here it is anyway: the Hausdorff dimension of a metric space (or say, a subset of $\mathbb{R}^n$ for some $n$) is the smallest possible $d > 0$ such that your set can be covered with balls of radius $r_1, r_2, \ldots$, where the sum $\sum_{i} r_i^d$ is finite. (Actually, rather than "smallest possible", it's the infimum, in the case that there is no smallest $d$.)
Under this definition, it turns out that a point has Hausdorff dimension $0$ and a line has Hausdorff dimension $1$. How is this possible? Well, in your question, you seem to be assuming some principle like

The union of any collection of sets of dimension $d$ must have dimension $d$. 

Assuming "dimension" = "Hausdorff dimension", this is true for a finite union. It is even true for a countably infinite union. But a line is composed of uncountably many points, and in the case of an uncountable collection the principle above is simply false.
