To help visualize things you can think of a vector as a directed line segment (a line with an arrow at one end to donate the direction), when you take the difference of 2 points it gives you a vector, take the maginitude/norm of that vector and it gives you the length of that line.

A simple example:

^||        * B (8, 4)|      |    |   * A (3, 1)    +----------------->__AB  = B - A = (8 - 3, 4 - 1)T = (5, 3)T__BA  = A - B = (3 - 8, 1 - 4)T = (-5, -1)T                    __                               __A and B are points, AB is the vector from A to B and BA is the vector from B to A.

Hope this helps,
Pat.

[edit]
If only one tuple of coordinates is provided, you can regard this either as a direction and scale or as a vector originating from (0,0). For example (5, 3)T alone can also be interpreted as the vector from (0,0) to (5, 3).

I also added the T as vectors are usually written vertically:

| x |  = (x, y)T| y |

[/edit]

say you have the vector (3,2), basically this is saying go three units to the right on the x axis and two units up on the y axis from where you are now.

There's no mention of points in this discription. However if we define one point as say the origin, then we can define other points with x and y coordinates. Now if we take the origin as (0,0) then if we walk in the direction of the vector (3,2) then we end up in point (3,2). The confusion is that they are represented by the same tuple of two numbers. If we take (1,1) as our starting point and we move 3 right and two up we end up in (4,3) which is different.

In this story the vector does not depend on the location of where we started.