Some partial answers taken from Neil Hindman, Dona Strauss, "Algebra in the Stone-Čech compactification", chapter 21 (aptly named "Other Semigroup Compactifications")
The most general is probably Theorem 21.31) :
- If $X$ is discrete, $Y$ compact, $g: \beta X \rightarrow Y$ continuous and onto, then $Y$ is isomorphic to a space of filters on $X$ (simply intersect the preimages )of points in $Y$)
- In particular, every compactification of $X$ can be viewed as a space of filters.
If you're interested in algebraic aspects, there's section 21.3 of the book.
If $(X,\cdot)$ is a semigroup (not necessarily discrete, but completely regular), "nice" semigroup compactifications such as the AP and WAP compactifications, i.e., the (maximal) topological and semitopological semigroup compactifications respectively, are very interesting objects. They also have nice descriptions as filters(see section 21.3 of the book).