Paragraphs and Essays share a common link: The paragraph is the building block of all essays. Paragraphs begin with a topic sentence that organizes the rest of the paragraph. The sentences following the topic sentence in the paragraph are called support sentences. These support sentences can be descriptive, explanatory, and/or analytical, depending upon the type of essay that is being written. A paragraph ends with a concluding sentence that wraps up the main ideas in the paragraph or acts as a lead-in to the next paragraph.

An essay consists of a set of organized paragraphs. The first paragraph in an essay is the introduction that contains a thesis sentence that acts much like a topic sentence in a paragraph; that is, the thesis organizes and states the main idea or purpose of the entire essay. The following paragraphs in an essay are support paragraphs that further explicate the thesis. These support paragraphs could be explanatory, descriptive, argumentative, cause/effect, etc. depending upon the type of essay being written. Lastly, an essay ends with a concluding paragraph that generally wraps up the main ideas in the essay, much like a concluding sentence in a paragraph.

The above information consists of the basic, guiding principles of paragraphs and essays.

This question is difficult to answer in one small space, but I will try.  Paragraphs usually deal with one idea only with a topic sentence to inform the reader what the paragraph is about.  The paragraph then explains, uses examples, and in general, uses the paragraph to prove the topic sentence.  An essay is larger than a single paragraph, the most simple being a five paragraph essay with an introduction paragraph which contains a thesis  statement, three body paragraphs, and a conclusion paragraph.  The thesis statement controls the entire essay by stating what the essay will prove or demonstrate. The three body paragraphs will be the three main points used to prove the thesis, and then the conclusion paragraph is used to restate the thesis, reiterate the three main points and summarize the conclusion.  No matter how long your essay is, whether five pages or five paragraphs, the idea is the same:  to state what you are going to prove and then do so in your essay using good grammar, varied sentence complexity, good mechanics, and logical, coherent arguments.