I'm sure a very simple question but I'm self-studying some homology and homotopy and this has suddenly started coming up without explanation. The first mention in my notes was for a map $\sigma :S^1 \rightarrow S^1 \vee S^1$ (where $\vee$ is the wedge)
$\sigma (t)= (2t,*) $ if $0 \le t \le \frac{1}{2}$
$\sigma (t)= (*,2t-1) $ if $\frac{1}{2} \le t \le 1$.
The next mention is the one given in the title of this question.
Thanks for your help in advance
It's the distinguished point in a pointed space. You know you're dealing with pointed spaces because the wedge sum operates on pointed spaces.
In your examples, from the context, it seems like the circle $S^1$ has been defined as $S^1=\mathbb R/\mathbb Z$, and the chosen point $*$ has been defined as ${*}=0=1$. That's the only way that $\sigma$ makes sense, such that $\sigma(0)=\sigma(1)$. The author could have written $0$ instead of $*$, but they probably decided that $*$ is better to communicate the point (har!) of the example.
Edit: And to be clear, a pointed space is a topological space that has been enriched with extra data. By "default", a topological space doesn't have a distinguished point, and $*$ doesn't denote anything in particular. So the pedantic answer to the question "What is the $*$ in topological maps?" would be: Nothing, it depends on the context.