The most commonly used method to represent a vector is with Cartesian coordinates. The units can be anything, but to start with, we could just consider the ``displacement vector'' which describes the difference in position of two points in space. You just plonk your vector on top of an x-y grid and read off the numbers on the x and y axes. Here's what I mean.
Click here to see the full figure
So this (two dimensional) vector, call it 
is represented by a pair
of numbers. The first one is the x component of the vector, 
, which
you get by reading
it off this figure. You can see that it's 3. The y component,
, can also be read of and is 5.5.
So you could write this vector as (3,5.5). If this vector
was three dimensional (which is more difficult to draw), then it
would be represented by three numbers.
But why did I place the grid the way that I did? Wouldn't I have been equally justified in plonking the vector on top of a grid going at some other angle, liked so?
Click here to see the full figure
Sure, that that seems fine too. After all this nothing better about one orientation rather than another. They're both equally valid coordinate systems. But when now I figure out the components of the vector, I see they're different they're different, (5.4, 2.3).
Which one is right? They're both right. You just have to be
clear what you're doing. Your specifying the components of
the vector with respect to a particular coordinate system.
So when you say 
, it's with respect
to a particular coordinate system that we plonked down.
Those numbers are meaningless unless you specify what
coordinate system you're using!