Stone–Čech compactificationof a compact space and the multiplier algebra of $C_0(X)$

Question

What is the Stone–Čech compactification $\beta X$ of a compact space $X$, is is $X$ itself? Or does it depend on the definition of compactification, whether the embedding $i:X\to \beta X$ is assumed to be dense or not? See for example here http://ncatlab.org/nlab/show/compactification. This question is similar with this question one-point compactification of a compact space . The background of my question is that in literature is always stated that the Multiplier algebra of $C_0(X)$, $M(C_0(X))$, where $X$ is a locally compact Hausdorff space $X$, is isomorphic to $C_b(X)$ and $C_b(X)$ can be identified with $C(\beta X)$,the continuous functions on the Stone–Čech compactification of $X$. But if $X$ is already compact, then it should be $C_b(X)=C(X)$ but if $\beta X\ncong X$ for compact $X$, you can't identify $C_b(X)$ with $C(\beta X)$ ... But I don't see the mistake here, but I think it depens on the embedding i/ the definition of compactification. Is it correct, or what is the problem here? Greetings