The definition in Wikipedia is:
The curve $\gamma: I \to \mathbb R^n$ $r$-times continuously differentiable with $I$ a non-empty interval of real numbers is regular of order $m$ (where $m\leq r)$ if, for every $t\in I,$
$$\{\gamma'(t), \gamma''(t),\cdots,\gamma^{(m)}(t)\}$$
is a linearly independent subset of $\mathbb R^n.$ In particular, a parametric $C^1$-curve $\gamma$ is regular iff $\gamma'(t)\neq \mathbf 0$ for any $t\in I.$
Does this imply that a particle moving through the curve constantly changes velocity? What is the motivation of this definition? What is the opposite of a regular curve?
If the $C^1$ curve $\gamma\colon I\to\mathbb R^n$ is regular, then by definition, $\gamma'\left(x\right)\ne\mathbf 0$ for all $x\in I$. This means that curve $\gamma$ never "slows down" to stop.
For example, constant curve $\gamma_1\left(t\right)=\mathrm{x}_0$ is not regular because $\gamma'_1=\mathbf 0$. On the other hand, straight line $\gamma_2\left(t\right)=\mathrm x_0 + t\cdot \mathrm x_1$ where $\mathrm x_1\ne \mathbf 0$ is regular because $\gamma_2'=\mathrm x_1\ne\mathbf 0$. This curve is not regular of order 2 (2-regular) because $\gamma_2''=\mathbf 0$. Circle, $\gamma_3\left(t\right)=\left(\cos t, \sin t\right)$, is regular of order 2. Notice that all three curves are $C^{\infty}$.