an ordered pair
An ordered pair is just two numbers written in a fixed order inside parentheses, like (3, 2) — but the word 'ordered' carries the whole point. It is the address of a single point on the Cartesian plane, and like a street address, the order of the parts is not negotiable: 'Apartment 3, Floor 2' is not the same as 'Apartment 2, Floor 3'.
Written (x, y), the first entry x is the x-coordinate (also called the abscissa) telling you how far to move along the horizontal axis, and the second entry y is the y-coordinate (the ordinate) telling you how far along the vertical axis. So (3, 2) means: go 3 right, then 2 up. Two ordered pairs are equal exactly when their first entries match and their second entries match — that is, (a, b) = (c, d) means a = c and b = d, both conditions at once. This is what makes the pair a faithful label.
Ordered pairs are the atoms of coordinate geometry: a curve is nothing but the whole collection of ordered pairs (x, y) that satisfy some equation, and a relation or function is described by which pairs it contains. The same idea scales up — three numbers (x, y, z) for a point in space, n numbers for a point in n-dimensional space — but on the plane it is exactly two, kept in order.
The pair (-2, 5) names the point 2 units left of the origin and 5 units up. Swapping to (5, -2) names a completely different point: 5 right and 2 down. Same two numbers, opposite corners of the plane.
'Ordered' means the slots have fixed jobs — x first, y second.
An ordered pair (a, b) is different from a two-element set {a, b}: a set ignores order and {a, b} = {b, a}, but the whole purpose of an ordered pair is that (a, b) is generally not (b, a).