Is it possible to add vectors that don't have the same initial point or terminal point? I have read in couple of books that describe addition of vectors not acting at the same point.
They describe it by saying we have to bring the initial point of one vector to the other and then resolve them into their respective x and y components and add the respective components to get the resultant.How is it that we can pull one vector to another point and still get the same effect?This seems physically impossible and very unintuitive. How is possible?
 A: A finite directed line segment (in the plane, say, for simplicity) has a starting point, a direction (slope) and a length.
A vector is not a directed line segment.
Rather, a vector is the class of all the directed line segments having the same direction and length. Each individual directed line segment in the class is a representative of the vector. 
In other words for any given vector $v$, there is a representative of $v$ at every point of the plane. 
We are always free to choose the most convenient representative of a vector. If we want to add one vector to another, we are free to choose a representative of the first vector based at the origin; and a representative of the second vector based at the head (pointy end, if you think of it that way) of the first; and then draw the diagonal as a representative of the vector sum.
Given a vector, we are not "moving" it to a different location. The vector already has a representative at every point. All we are doing is choosing the most convenient representative for the given problem.
Also see Problems with the definition of vectors as directed line segments in $\mathbb{R}^3$.
