Let $X$ be a topological space and $I$ an interval in $\mathbb{R}$. A continuous curve in $X$ is a continuous map $\gamma : I \to X$.
Let $X$ be a smooth manifold and again, let $I$ be an interval in $\mathbb{R}$. A smooth curve in $X$ is a smooth map $\gamma : I \to X$.
Note, it both cases, a curve is more than its image. That is, given two curves $\gamma_1 : I_1 \to X$ and $\gamma_2 : I_2 \to X$, it may be the case that $\gamma_1(I_1) = \gamma_2(I_2)$. A particular instance of this occurs when there is a map $\sigma : I_2 \to I_1$ which is a homeomorphism in the case of continuous curves or a diffeomorphism in the case of smooth curves, such that $\gamma_2 = \gamma_1\circ\sigma$. In this case, we say that $\gamma_2$ is a reparameterisation of $\gamma_1$.