Depending on the type of essay you are writing, requirements can change significantly. Originally, essays were opinion pieces, similar to editorials today. For the purposes of this answer, we will assume the essay is for a school or college.
The first step in an essay is the Outline, which is optional. This is a numbered list of all the points you want to make in your essay, with subtopics in their appropriate spots. The outline contains within itself all the other parts of the essay, but in short, unexplained form. The outline may or may not be part of the final essay. It is often a good idea to include it, whether as a "table of contents" or as an endnote.
The first major part of the essay is the Introduction. This explains the motive behind the paper. The introduction can be as long as one page and is usually kept to three paragraphs maximum. You should be careful not to use the introduction to argue any of your points. The introduction should be as neutral and clear as possible.
The next important part is the Thesis. This is a clear statement of the theme or conclusion you hope to reach through the essay. The thesis can be a simple sentence, but should be no longer than one paragraph. The thesis usually comes after your introduction and leads directly into the main body of the essay.
The argument part of the essay consists of the Body. The body is composed of paragraphs, each of which explains and argues one point, usually transitioning into the next with the last sentence. The body can be as long or as short as you want, as long as you touch on all the points mentioned in the introduction and thesis. The body for an academic paper is usually two-to-three pages long.
The next part is the Conclusion. This is where you sum up all the points you have made and explain the results. The conclusion is usually no more than two or three paragraphs long; many are one paragraph. There should be no argument of points in the conclusion.
The final part of an essay is the Bibliography. This is a vital part of the essy, especially in schoolwork, where you cite any sources quoted or drawn from. The important thing to remember about citation is that there are several accepted formats to be used, each for a specific audience. Legal essays cite differently than medical essays, for example. The bibliography is absolutely essential if you are making arguments from outside sources; without a bibliography, you may risk plagiarism.
The critical essay is an important part of any school program. Using the proper parts correctly will ensure that your essay cannot be faulted on technical levels.