the unit tangent vector
Stand on a winding road and point straight ahead in the direction you are facing, taking exactly one step's worth of arrow no matter how fast you are moving. That arrow is the unit tangent vector. It captures the pure direction of travel along a curve, with all information about speed deliberately stripped out, so that two travellers on the same road at the same point share the same arrow even if one is sprinting and the other strolling.
Precisely, for a regular parametrized curve r(t) the velocity r'(t) points in the direction of motion but its length is the speed. To keep only the direction, divide by the length: T = r'(t) / |r'(t)|. This makes T a unit vector, |T| = 1, pointing along the tangent line to the curve at that point. If the curve is parametrized by arc length, the speed is already 1, so the formula simplifies beautifully to T = r'(s) — taking the derivative with respect to arc length automatically gives a unit vector. The tangent line at the point is the straight line through that point in the direction of T; it is the best straight-line approximation to the curve there, the path you would fly off along if the curve suddenly vanished.
T is the first member of the moving frame that travels with the bug. As you move along the curve, T turns; the rate at which it turns is precisely the curvature, and the direction in which it turns gives the principal normal. So the unit tangent is the seed from which the whole Frenet-Serret apparatus grows. One honest subtlety: T depends on the orientation of the curve. If you reparametrize so as to travel the same road backwards, every T reverses to -T, because 'forward' has changed meaning — direction is genuine data that you must choose and keep.
For r(t) = (cos t, sin t), r'(t) = (-sin t, cos t), and |r'(t)| = 1, so T(t) = (-sin t, cos t). At t = 0 the point is (1, 0) and T = (0, 1): you are at the rightmost point of the circle and moving straight up, exactly along the tangent line x = 1.
T = velocity divided by speed: pure direction, length 1, tangent to the curve.
T records direction, not speed: |T| is always 1. And T points whichever way you traverse the curve — reversing the parametrization sends T to -T, so the tangent vector carries an orientation while the tangent line does not.