This has been an extremely hot area of research for the past several
decades, not to mention a couple hundred years of research in
embryology.  It has practical implications in stem cell research, which
may lead to developments in repairing or replacing damaged organs, and
in understanding carcinogenesis, since cancerous cells express genes
that are silent in normal cells of their original kind.

AFAIK, there's no simple answer.  There are many different mechanisms,
and many are active in the same cells.  Multicellular organisms have
had about 750 million years to come up with methods and elaborate them,
resulting in amazingly baroque systems that none the less work, most of
the time.  This is typical of living systems -- elaborate kludges full
of props, fail-usually-safes, unexpected interactions, bits of things
developed from unlikely origins, special cases and exceptions.  They
usually work remarkably well in most cases, under most conditions, for
a while at least.

When the "Central Dogma" of DNA -> RNA -> protein was developed fifty
years ago, it was hoped that the mysteries of development and
differentiation would also have an underlying elegant simplicity.  This
appears not to be the case, and the "Central Dogma" itself only applies
most of the time anyway, alas!