a regular curve
When the bug walking along the wire never comes to a dead stop, its motion is smooth and well-behaved, and at every instant it has a clear direction of travel. A regular curve is a parametrized curve that satisfies exactly this no-stopping condition. The point of the word 'regular' is to rule out the bad moments where the machinery of tangents, normals and curvature would jam.
Precisely, a parametrized curve r(t) is regular if its velocity vector never vanishes: r'(t) is not the zero vector for any allowed t, equivalently |r'(t)| > 0 everywhere. Why does this matter? Because the direction of travel is the direction of r'(t), and you cannot point in 'the direction of the zero vector'. At a point where r'(t) = 0 the curve can stop and reverse, forming a sharp corner called a cusp even when the formulas x(t), y(t) are perfectly smooth. The classic example is r(t) = (t^2, t^3): here r'(t) = (2t, 3t^2), which is (0, 0) at t = 0, and indeed the curve has a sharp cusp at the origin. Regularity is the clean condition that guarantees a well-defined tangent line at every point.
It is worth being honest about the two layers here. Smoothness (being able to differentiate the coordinate functions) is about the formulas; regularity (the velocity never being zero) is an extra requirement about the speed. A curve can be perfectly smooth as a set of formulas yet fail to be regular, and then it may have corners. Once a curve is regular, you are free to reparametrize it by arc length and build the whole Frenet-Serret apparatus on it; regularity is the entry ticket to the entire local theory of curves.
r(t) = (t^2, t^3) is smooth (both coordinates are polynomials) but not regular: at t = 0 the velocity r'(0) = (0, 0). The image has a sharp cusp at the origin, where no single tangent direction exists. By contrast r(t) = (cos t, sin t) has |r'(t)| = 1 everywhere, so it is regular and has a clean tangent at every point.
Smooth formulas can still hide a cusp; regularity (nonzero velocity) forbids it.
Regular is not the same as smooth. Smooth means the coordinate functions are differentiable; regular adds that the velocity is never zero. Only regularity guarantees a tangent line and lets you reparametrize by arc length.