How should I think of schemes of the form $\operatorname{spec} k[f_1, \ldots, f_n]$, $f_i \in k[x_1, \ldots, x_m]$? What of the map on spectra induced by the inclusion $i : k[f_1, \ldots, f_n] \to k[x_1, \ldots, x_m]$? What are the processes that construct these kinds of spectra?
What I am looking for: a reasonable answer to thinking of schemes of the form $k[x_1, \ldots, x_n] / I$ is as closed subschemes, varieties in $k^n$, etc.... is there an analogous familiar object for the spectra of subrings?
I know (vaguely) that sometimes these are quotients by group actions (the ring of invariants). I also know (vaguely) that sometimes they are obtained by gluing along closed subschemes. Are there other process that can construct schemes like these? There seems to be a common theme here, for instance we are taking $A^n$ and pinching, twisting, covering it onto itself - except that I think the dimension can go up, weirdly, as in $k[x,xy,xy^2,\ldots]$.)
Can this be made more precise?