In the context of proper time "extremal" mostly means maximal in practice. And proper time is what you measure with a wristwatch, or by aging. So you can think of "extremal proper time" as "maximal aging", usually.
However, there is a significant difference between "mostly means in practice" and "always means".
To understand more precisely the definition of extremal, first consider a function of a single variable, y = f(x)
An an extremal point, dy/dx = 0, i.e. the slope is horizontal.
Extremal points can be either a maximum, a minimum, or a saddle points.
If all this isn't a review, or if you lack calculus, you may need to do some further reading and research to fully understand this. (I'm sorry, but I don't know your background).
A quick example might help. Rather than draw graphs, which is the clearest, I'll use some well known simple equations:
y=x^2 has a minimum at x=0
y = -x^2 has a maximum at x=0
y = x^3 has a saddle point at x=0
We can apply similar definitions to functions of more than one variable, in which case we require all the derivatives to vanish to have an extremal point.
In the case of an extremal path, we do a bit of fancy mathematical footwork to extend the same basic definition we use with one variable to an infinite number of variables.
