As described here (and as I always thought was the most general definition of boundary), a possible definition of the boundary of a subset $S$ of a topological space $X$ is $\partial S = \overline S \backslash \mathrm{int}(S)$.
I am reading Allen Hatcher's book on algebraic topology, and they often refer to "the" boundary of a topological space, for instance by saying that $\partial D^2 = S^1$. How is this notion of boundary uniquely defined? Because if I take an injection of $D^2$ into $\mathbb R^2$, I'm going to get $S^1$, but if I take the injection of $D^2$ into $\mathbb R^3$, then $D^2$ becomes its own boundary. I know that the $\partial$ is an important operation in topology so I'm trying to figure it out, and I feel like it doesn't quite make sense here.
So can anyone describe precisely what is meant by $\partial X$ when $X$ is a topological space? Or at least explain what is meant by that in particular contexts where it is used.
The notion of boundary that you are looking for comes from the definition of topological manifolds with boundary. As opposed to a regular manifold $X$, a manifold with boundary has the property that each point in $X$ has an open neighborhood which is homeomorphic to an open set in the euclidean half space $\mathbb{R}_+^n=\{(x_1,\dots,x_n)\in\mathbb{R^n}:x_n\ge0\}$. Thus we then define $\partial X$ to be the points which when mapped to $\mathbb{R}_+^n$ have $x_n=0$.
This definition has the benefit that an embedding of $X$ into some other space does not change $\partial X$. Thus $\partial D^2=S^1$ irregardless of whether you view it as living in $\mathbb{R}^2$ or in $\mathbb{R}^3$.