A smooth curve $\gamma:[a,b] \rightarrow \mathbb{C}$ is a continuously differentiable map such that $\gamma'(t) \not= 0$ for all $t \in [a,b]$. A contour is a curve that is equivalent (up to continuous reparameteriztion) of a concatenation of finitely many smooth curves.
A simple smooth curve, $\gamma:[a,b]\rightarrow \mathbb{C}$, is a smooth curve that is simple. i.e. that $\gamma(t) \not = \gamma(s)$ for all $s \not= t \in[a,b]$ except possibly when $s=a, t=b$.
Exercise 25: A curve is a contour iff it is the concatenation of finitely many simple smooth curve.
EDIT: I came back to the problem but still did not proceed far for =>. I could deduce:
$\gamma$ does not take the same value infinitely many times: Take an infinite subsequence $\{t_n\}$, $\gamma(t_i)=\gamma(t_j)$ for all $i,j$, then as $[a,b]$ is comapct, exists converging subsequence, $\gamma_{t_j} \rightarrow \alpha$. Then $\gamma'(\alpha)=0$.
Any hints?
It is enough to show that every smooth curve $\gamma: [0,1] \rightarrow \mathbb C$ can be written as concatenation of finitely many simple smooth curves. Clearly, there exist smooth curves $\gamma_{\mathfrak R}: [0,1] \rightarrow \mathbb R$ and $\gamma_{\mathfrak I}:[0,1] \rightarrow \mathbb R$ such that $\gamma(t) = \gamma_{\mathfrak R}(t) + i\gamma_{\mathfrak I}(t)$. Since $\gamma'(t) \neq 0$, it follows that $\gamma_{\mathfrak R}'(t) \neq 0$, and we can use inverse function theorem from real analysis. Therefore, for every $t \in [0,1]$, there exists $\varepsilon_t > 0$ such that $\gamma_{\mathfrak R}(s)$ is injective for every $s \in I_t = (t-\varepsilon_t, t+\varepsilon_t)\cap[0,1]$. Sets $I_t$ are open cover of $[0,1]$ (in topology of $[0,1]$), and therefore, since $[0,1]$ is compact, there exists finite subcover $\{I_{t_1}, I_{t_2}, \ldots, I_{t_N}$} of $[0,1]$ with open sets. Let $0 = a_0 < a_1 < \ldots < a_k=1$ be partition of interval $[0,1]$ such that each $[a_i, a_{i+1}]$ is subset of some $I_{t_j}$ and you've got desired simple smooth curves $\gamma_i = \gamma \restriction{[a_i, a_{i+1}]}, \quad i \in \{0,1, \ldots, k-1\}$ which concatenate to curve $\gamma$.