The first is between a's essence and a. a's essence is identical with a. a is identical with a's essence. No matter how you slice it, they're the same thing. (a's essence encompasses all of what a is.) Essence and nature are synonyms.
The second is between a's essence and its substantial form (natural kind). a's essence is what a is; a's substantial form is the natural kind-universal or -instance that is a constituent of a. “a's essence” (a) includes its particularity, and we more or less come to know it through the senses; a's substantial form doesn't include a's particularity, and has to be abstracted from that which we acquire through the senses. If you muddle this distinction, you're going to run into questions like “How do we know x individual's essence?”
The third is between a's essence and a's existence. a's essence is just a. a's existence (say Thomists) is something additional to a (e.g. Barry Miller's existence property-instances). The result (if they're right) is that there is a real distinction between a and existing a.
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