I have seen the plot below in many textbooks before. What do the X axis and the Y axis represent exactly? What does it mean for the Hinge loss to "touch" the 0-1 loss on the (x,y) pair (0,1)?
If helpful, the source of this picture is from the entry on Hinge Loss on Wikipedia.
The x-axis is the score output from a classifier, often interpreted as the estimated/predicted log-odds. The y-axis is the loss for a single datapoint with true label $y = 1$.
In notation, if we denote the score output from the classifier as $\hat s$, the plots are the graphs of the functions:
$$ f(\hat s) = \text{Zero-One-Loss}(\hat s, 1) $$ $$ f(\hat s) = \text{Hinge-Loss}(\hat s, 1) $$
