It's a complicated answer. More than 200 cell types, each type of cell inherits a unique expression of receptors, internal and external. Diffusion of signals through the plasma membrane and/or nuclear compartment may act directly as cofactors, activators, etc. The specific sequestration, and pattern of expression of external receptors also influence what signals become bound and relayed. The cell will also have a motif of internal/cytoplasmic/nuclear proteins expressed that complex or become directly activated by interaction with any of the above, etc. So the currently expressed set of internal, and external proteins determine not only how the cell interprets the signal, but also how they respond.
As for the actual genetic material, you'll find inherited motifs like DNA methylation, histone acetylation patterns etc. influence what parts of the DNA can actually be accessed. Per the above schema of signaling pathways, the signals received by each individual cell differ based on the actual cell type. In any case there are patterns of activators/repressors, enhancers/silencers that are switched on or off by these signals. Some journals are also introducing evidence insoluble receptors like EGFR+ligand can actually internalize and translocate to the nucleus and act as transcription regulators.
The final note, is that due to alternative splicing and transcription of proteins that promote or repress splicing sites, in a combination with the above concepts a cell can regulate for thousands of genes. This is a very general answer, however. The concept to take home, though, is that differentiated cells have their own specific motifs they express that make "reading" different signals possible, and also makes differing responses to what could be the same signal possible. These motifs also allow for the accurate expression of the relevant portions of the genome (not all genes are expressed at all times in all cells).