You're not going to find a uniform usage. The best you can do, probably, is to fix upon particular schools of thought and see how they used the terms.
You can find all of these terms in Antiquity, and I'd venture to guess that most of them were at least popularized by Aristotle. For more on them, read the SEP entry on Aristotle's metaphysics.
Here is my understanding of how these terms are used in contemporary (analytic) philosophy. I will include references where they come to mind:
Substance: the fundamental kinds of things. It comes up in Philosophy of Mind, for instance, when we speak of Substance Monists (who believe in only a single sort of substance, most commonly physical substance) or Substance Dualists (who believe in two sorts of substance, most commonly mental and physical substance; Descartes is commonly taken to be a substance dualist).
Essence: often brought up in discussion of essential properties see Plantinga's Nature of Necessity for a contemporary discussion of essences. Kit Fine also has some articles on essentialism. An essential property is (at least) one that an object possesses necessarily. It will sometimes be said to be definitional of that object. To give a plausible example, it is an essential property of a chair that it be able to be sat upon.
Existence: I've never really seen this defined, and contemporary analytic metaphysics tends to make no distinction between existing and being. The Meinongians famously distinguished between different kinds of being, where some things with being might not exist. Contemporary analytic metaphysicians tend to follow Quine in holding that there are no things that have being but do not exist. You might be interested in criteria for existence, in which case you could look at the literature on Ontological Commitment, which is the study of when a theory (or person) is committed to a certain kind of thing. Quine famously said "to be is to be the value of a (first-order) variable". See Quine's "On What There Is".
Entity: I am aware of no special usage of this term. As far as I know, an entity, a being, a thing, an object, etc. are all used more or less interchangeably, with the exception that "being" (and possibly "entity", depending on author) might be reserved for living things.