coordinate vector
Once you fix a basis — a chosen set of reference vectors — every vector in the space gets a tidy list of numbers called its coordinate vector. Those numbers are simply the weights you need in a linear combination of the basis vectors to rebuild the original vector. The basis is the 'measuring sticks', and the coordinate vector records how many of each stick you laid down to reach your target.
The crucial and often surprising point is that the very same arrow has different coordinates in different bases. The arrow does not move; only the rulers you measure it against change. The numbers (3, 1) that we casually write for a vector are really its coordinates in the standard basis (1, 0) and (0, 1) — we just rarely say so out loud. Switch to a different, slanted basis and that identical arrow earns a brand-new list of numbers.
A tiny example: in the standard basis, v = 3*(1, 0) + 1*(0, 1), so its coordinate vector is (3, 1). Now choose the basis b1 = (1, 1) and b2 = (1, -1). The same v can be written 2*b1 + 1*b2 = (3, 1), so in this new basis its coordinate vector is (2, 1). Same arrow, two different coordinate lists — which one you get depends entirely on the basis you chose.
One fixed arrow, two coordinate vectors — the list of numbers depends on which basis you measure against.
The numbers you write for a vector are always coordinates relative to some basis — usually the standard one, even when nobody says so.