a parametrized curve
Imagine a tiny bug walking along a wire, or the dot of a pen tracing a shape on paper. At every instant of time the bug is at one definite spot, and as time runs forward it sweeps out a path. A parametrized curve is exactly that idea made precise: a rule that, for each value of a single number (think of it as time), hands you a point. Instead of describing a curve all at once by an equation like x^2 + y^2 = 1, you describe it as a journey — where you are at each moment.
Precisely, a parametrized curve in the plane is a function r(t) = (x(t), y(t)) that takes a number t from some interval and returns a point; in space it is r(t) = (x(t), y(t), z(t)). The variable t is the parameter. For example r(t) = (cos t, sin t), with t from 0 to 2 pi, traces the unit circle: at t = 0 you are at (1, 0), at t = pi/2 you are at (0, 1), and so on counterclockwise. We usually ask the coordinate functions to be smooth, meaning you can differentiate them as often as you like, so that velocity and acceleration make sense. The derivative r'(t) = (x'(t), y'(t)) is the velocity vector: it points in the direction of travel and its length is the speed.
The crucial honest point is the difference between the curve (the set of points actually visited, the bent wire itself) and the parametrization (the particular schedule of how a point moves along it). One geometric shape can be walked in infinitely many ways — fast or slow, forwards or backwards, even pausing — so many different parametrized curves share the same image. Most of differential geometry of curves is the art of separating the genuine geometry of the shape from the accidental choices of the schedule, and a recurring theme is to find one canonical, fair schedule (the arc-length parametrization) in which the bug moves at constant unit speed.
The same parabola y = x^2 can be parametrized as r(t) = (t, t^2) for t in (-infinity, infinity), or differently as r(s) = (2s, 4s^2). Both trace exactly the same set of points, but at any given parameter value they sit at different places and move at different speeds — same road, different driving.
One image, many parametrizations: the shape is the road, the parametrization is the trip.
Do not confuse the curve with its parametrization. r(t) = (cos t, sin t) and r(t) = (cos 2t, sin 2t) both draw the unit circle, but the second goes around twice as fast — they are different parametrized curves with the same image.