Peter Johnstone's book Stone Spaces (p. 144) proves that for any X, maximal ideals in C(X) are the same as maximal ideals in C_b(X) (bounded functions), i.e. the Stone-Cech compactification \beta X. Indeed, if I is a maximal ideal, let Z(I) be the set of all zero sets of elements of I; this is a filter on the lattice of all closed sets that are zero sets of functions. Then I is contained in J(Z(I)), the set of functions whose zero sets are in Z(I), so by maximality they are equal. But also, by maximality, Z(I) must be a maximal filter on the lattice of zero sets, and we get a bijection between maximal filters of zero sets and maximal ideals in C(X). Now the exact same discussion applies to C_b(X) to give a bijection between maximal filters of zero sets and maximal ideals of C(X) (since the possible zero sets of bounded functions are the same as the possible zero sets of all functions). But the maximal ideals of C_b(X) are just \beta X.

The difference between C_b(X) and C(X) is that for C_b(X), the residue fields for all of these maximal ideals are just C, while for C(X) you can get more exotic things. Indeed, if a maximal ideal in C(X) has residue field C, then every function on X must automatically extend continuously to the corresponding point of \beta X. This can actually happen for noncompact X, e.g. the ordinal \omega_1.

Section IV.3 of Johnstone's book has a pretty thorough discussion of this stuff if you want more details.

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Peter Johnstone's book Stone Spaces (p. 144) proves that for any X, maximal ideals in C(X) are the same as maximal ideals in C_b(X) (bounded functions), i.e. the Stone-Cech compactification \beta X. Indeed, if I is a maximal ideal, let Z(I) be the set of all zero sets of elements of I; this is a filter on the lattice of all closed sets that are zero sets of functions. Then I is contained in J(Z(I)), the set of functions whose zero sets are in Z(I), so by maximality they are equal. But also, by maximality, Z(I) must be a maximal filter on the lattice of zero sets, and we get a bijection between maximal filters of zero sets and maximal ideals in C(X). Now the exact same discussion applies to C_b(X) to give a bijection between maximal filters of zero sets and maximal ideals of C(X) (since the possible zero sets of bounded functions are the same as the possible zero sets of all functions). But the maximal ideals of C_b(X) are just \beta X.

The difference between C_b(X) and C(X) is that for C_b(X), the residue fields for all of these maximal ideals are just C, while for C(X) you can get more exotic things. Indeed, if a maximal ideal in C(X) has residue field C, then every function on X must automatically extend continuously to the corresponding point of \beta X. This can actually happen for noncompact X, e.g. the ordinal \omega_1.

Section IV.3 of Johnstone's book has a pretty thorough discussion of this stuff if you want more details.