TITLE: Stone–Čech compactification of real line
QUESTION [14 upvotes]: I know that $[0,1]$ and a unit circle $\mathbb{S}^1$ are one-point compactifications of $\mathbb{R}$ under some suitable homeomorphism. But how does one construct the Stone–Čech compactification?

REPLY [9 votes]: The Stone-Čech compactification of $\mathbb{R}$ can be functorially built as the maximal spectrum of $C_b(\mathbb{R})$, the ring of bounded continuous real functions on $\mathbb{R}$.
The maximal spectrum $\operatorname{Max}(R)$ of a commutative ring is the set of all maximal ideals, with the spectral topology, where a basis of closed sets is given by
$$
V(I)=\{\mathfrak{m}\in\operatorname{Max}(R):\mathfrak{m}\supseteq I\}
$$
where $I$ is any ideal of $R$. In general this topology is not Hausdorff, but it can be proved that if $X$ is a completely regular Hausdorff space, then the space $\beta X=\operatorname{Max}(C_b(X))$ is compact Hausdorff.
Suppose $f\colon X\to Y$ is a continuous map, where $Y$ is completely regular Hausdorff. Then we have an induced ring homomorphism $f^*\colon C_b(Y)\to C_b(X)$, mapping $\varphi\in C_b(Y)$ to $\varphi\circ f$. If $\mathfrak{m}$ is a maximal ideal of $C_b(X)$, then $\beta f(\mathfrak{m})=(f^*)^{-1}(\mathfrak{m})$ is a maximal ideal of $C_b(Y)$. It's easy to show that $\beta f$ is continuous under the spectral topology, so we get a continuous function $\beta f\colon\beta X\to\beta Y$.
Moreover we get an embedding $X\to \beta X$ by mapping $x\to\mathfrak{m}_x$ defined by
$$
\mathfrak{m}_x=\{\varphi\in C_b(X):\varphi(y)=0\}
$$
In the special case when $Y$ is compact, the maximal ideals of $C_b(Y)=C(Y)$ are all of the form $\mathfrak{m}_y$, for a unique $y\in Y$, so $Y$ can be identified with $\beta Y$. This shows that $\beta X$ indeed satisfies the universal property of the Stone-Čech compactification, that is, every continuous map $X\to Y$, where $Y$ is compact Hausdorff, can be extended to a continuous map $\beta X\to Y$.
You should be able to find this construction of the Stone-Čech compactification in several books on topology, I believe it is also in Kelley's book.
There's no “explicit” description, because the ideals of $C_b(X)$ can be described in terms of nonprincipal ultrafilters, whose existence depends on the axiom of choice. Only some cases admit an explicit description, because of special properties of $X$.