Motivation of Stone–Čech compactification What are some uses of Stone–Čech compactification? What is the motivation for introducing this notion?
Most textbooks on topology construct Stone–Čech compactification and prove that this construction satisfies a particular universal property (which essentially sais that it is the left adjoint to the inclusion functor from the category of compact Hausdorff to the category of topological spaces).
But this seems to be quite isolated from the rest of (algebraic) topology. For instance, Munkres' book on topology has a short section on the Stone–Čech compactification, but after that section he doesn't use the Stone–Čech compactification anywhere else in the book, as far as I can see.
Wikipedia gives one application of the Stone–Čech compactification: it can be used to construct the dual space of the bounded sequences of reals.
I wonder: That's all??! Probably not! What's the real motivation and use of Stone–Čech compactification?
 A: In my opinion the motivations and the applications for a notion do not always have to be connected. In this case I think the motivation is fairly clear: the fact that there is a universal way to compactify any topological space has a clear conceptual interest. Independantly of any application, this is a valuable fact, which can be situated in a general theoretical context (for example, it can be seen as the fact that the category of compact spaces is a reflexive subcategory of topological spaces).
Now the fact that most textbooks do not spend a lot of time on it shows that indeed, this is not necessarily a construction that is used a lot in practice (or at least not as much as a lot of other basic topological tools). One of the reasons is that this compactification tends to be a huge and messy space, so for instance in algebraic topology, where we usually want to work with "nice" spaces (such as CW-complexes), this construction will in general not be very useful.
It does not mean that there are no applications of this construction (people have already given a few in the answers and comments), but in my opinion they are not the reason why it is generally introduced.
A: Condensed mathematics is another recent "application" of Stone-Cech compactifications.  In the last several years, Clausen and Scholze (and also Barwick and Haine) have been developing a way of doing algebra and topology together based on "condensed" objects, i.e., sheaves on a site of profinite sets.  For example, condensed sets should be thought of as a substitute for topological spaces; there is a Yoneda embedding of spaces into condensed sets.  This perspective remedies defects in the usual category of say topological abelian groups, which do not form an abelian category.  However, the category of condensed abelian groups do form an abelian category satisfying a bunch of extra nice properties (Grothendieck's axioms), and the reason they do so is because of the existence of compact projective generators which can be functorially generated using Stone-Cech compactifications.
A: I'd say the original motivation was to show that every Tychonoff space $X$ had a compactication (a compact Hausdorff space in which $X$ is densely embedded) and the natural construction that people came up with (via Tychonoff cubes $[0,1]^I$) turned out to have very nice special properties: this compactification is the maximal one for $X$ (all other conceivable compactifications of $X$ are quotients of it), has nice function extension properties and so had (as we'd now say) functoriality.
Of course for discrete spaces it occurs as a Boolean space for power set algebras (another natural road) and so is part of nice theory and duality via an algebraic road.
It also occurred naturally when people started studying rings of continuous functions $C(X)$ for Tychonoff spaces, its points correspond to the maximal ideals of such rings.
There are also applications: van der Waerden's theorem can be proved by considering the compact semigroup $\beta \Bbb N$. And as $\ell^\infty \simeq C(\beta \Bbb N)$ it allows for a slightly more concrete representation of the dual of $\ell^\infty$ as a space of measures on a compact space $\beta \Bbb N$, which is an interesting space in its own right (see the Handbook of General Topology which has a whole chapter on this one space).
