Does this mean that a hyperplane in four dimensional space is defined by any four points?
Yes, if they are not all in the same 2-dimensional plane.
(The same way that 2 point determine a line if they're not equal, and 3 points determine a 2-dimensional plane if they're not all in a line.)
In general, in a space of dimension at least $n$: if you pick $n+1$ points in the space, they will determine a unique "plane" of dimension $n$, unless they're all in the same "plane" of dimension $n-1$.
Can the concept be applied to higher dimensions and still make sense or be useful?
Yes. These things are typically taught in a course in Linear algebra. In linear algebraic terms, an $n$-dimensional "plane", as I was referring to it, is called an $n$-dimensional affine subspace.
Linear algebra is arguably one of the most fundamental pieces of math in use today, so it should certainly be considered useful.