I'm having trouble interepreting the following notation on the wikipedia for martingales:
$$\mathbf{E} ( Y_{t} \mid \{ X_{\tau}, \tau \leq s \} ) = Y_s\quad \forall s \le t$$
The stopping time of the conidtioning variable is $\tau$. If the process $(X_t)$ stops at $\tau$, shouldn't we have the following instead? -- $$\mathbf{E} ( Y_{t} \mid \{ X_{\tau}, \tau \leq s \} ) = Y_{\tau}\quad \forall s \le t$$
No. Note that $\tau$ is not a stopping time, it is just an arbitrary real number less than or equal to $s$.
In particular, this property is the same as saying that for all $s \le t$,
$$ \mathbb E [Y_t \vert \{X_u,u\le s \}] = Y_s. $$
The conditioning means that our information is generated by the path of the process $X$ up to time $s$. (Compare this with the definition of the property in discrete time.)