Essence in this case refers to the ancient philosophical idea (most closely associated with Plato) that all things have a predefined, ideal set of characteristics. For instance, the Essence of a chair is that it has four legs, a back, and people sit on it.
However, not everything matches its Essence. You might have a chair with three legs, or a broken back, or that no one sits on. The actual details of a particular chair make up its Existence.
The idea that Existence precedes Essence is that –for human beings –there is no predefined pattern that we must fit into. We live our lives, and that in turn defines what we truly are, not any idealized set of characteristics. This idea is the heart of Sartre’s version of Existentialism.
The implications are that we must create our own meaning, place our own value on our acts, and that our individual freedom is absolute and unbounded.
As a side note, Sartre, although an atheist, gave what I consider to be one of the best ever descriptions of God, as the “Union of Existence and Essence”, meaning that God is the full Existential realization of every perfect, ideal or Essential attribute of God. Sartre, of course, described that as an impossibility, but it is also a good description of what a believer believes God to be.