Let $\Omega^{p,q}(M)$ be the $C^{\infty}$ $(p,q)$-forms. One always has a double complex with $\Omega^{p,q}(M)$ in position $(p,q)$. The cohomology in the $q$ direction is Dolbeault cohomology, the cohomology of the total complex is deRham cohomology. (In each case, essentially by definition.) Whenever you have a double complex, you get a spectral sequence. On the first page of the spectral sequence is Dolbeault cohomology; the spectral sequence converges to deRham cohomology. This is sometimes called the Hodge-de Rham spectral sequence, and sometimes called the Frolicher spectral sequence.
If $M$ is compact Kahler (in particular projective) then the spectral sequence collapses at the first page; all maps between Dolbeault groups are zero. So $H^k(M) \cong \bigoplus_{p+q=k} H^{p,q}(M)$ in this case.
If $M$ is Stein (in particular, affine) then the only nonzero Dolbeault groups are the $H^{p,0}(M)$, corresponding to holmorphic $p$-forms. The next page takes the cohomology of the complex of holomorphic $p$-forms; I'll term this ``holomorphic deRham". After that, there are no further maps, so holomorphic deRham equals deRham.
In general, the spectral sequence can be arbitrarily nondegeneratecan be arbitrarily nondegenerate.