An essay is a formal writing sample of a given topic composed of multiple paragraphs.  The largest difference between the two types of writing is that the essay usually involves more preparation work.  Traditional essays involve five paragraphs, in which the first serves as an introduction and includes a thesis statement- a critical idea that defines the purpose of the paper.  The subsequent paragraphs develop and explore the thesis through the use of evidence and analysis in its explanation . The concluding paragraph summarizes the findings that have been presented in the essay and has proven the thesis statement in a lucid and reasonable manner.  There will be disagreements on the outline of it, but traditionally this is how an essay breaks down.

By contrast, a paragraph is shorter and by virtue of this does not have as many requirements.  There is a topic sentence to open the paragraph that defines its purpose and the remaining sentences in the paragraph serve to support and enhance that topic sentence.  If you want to think of it in a comparative way, the topic sentence is the thesis statement of a paragraph.  The development of a paragraph can lead to an essay, and normally, this is how good essays begin.  Different teachers have different belief systems on it, but roughly, I think this is what is meant by the difference between an "essay" and a "paragraph."