The posts above all give you good advice especially post #9 being so specific about structuring each part of your essay.  Good mechanics are expected in  an essay, so check the grammar etc of your paper.  One addition I would make is that when you write an essay and you have finished all of the above, go back to your paper and revise.  First, cross out the "there is, there are, this is" beginnings to sentences and look for the subject in the words following those phrases.  Look for how many of your sentences start with the word I or two or three letter words.  Can you rephrase or combine the sentences to cut down on the number of times you use "I"?  Think about using compound or complex sentences to enhance your writing.  Use what you have learned about writing which maybe you forgot to utilize.

A text that is often used in high schools and colleges is Bailey and Powell's The Practical Writer. This most useful text suggests that a standard essay is composed of five paragraphs: Introduction, 3 Body Paragraphs, and a Concluding Paragraph.

1. The Introductory Paragraph is composed of what is called a "Motivator," also known as a "hook," especially in journalistic writing.  This motivator is an anecdote, a reflection or observation, a relevant quote, or a question. From this motivator, the writer leads into the thesis, a general statement that expresses the main idea of the essay, the idea that the entire essay should support. This thesis is followed by a "blueprint," a list of the ideas that will become the 3 topic sentences for the body paragraphs. These are essentially the 3 opinions that the writer will support. Thus, the introduction (a) grabs the reader's attention and (b) informs the reader of the main idea of the essay and how it will be developed.
2. The 3 Central paragraphs are formed from the 3 opinions.  Each paragraph has a topic sentence, supporting sentences which contain reasons, arguments, counter-arguments, examples, citations from a text, testimonials, facts, statistics, or narrative examples. Which type of support a writer uses depends upon the writer's purpose. The paragraph, then, ends with a concluding sentence which will make also a transition to the next paragraph.  Thus, the three basic parts of the paragraphs are (a) a topic sentence, (b) specific support, and (c) a reworded topic sentence.
3. The Concluding paragraph rewords the thesis, reminding the reader of this main point.  The writer can begin the concluding paragraph by restating the three opinions. For example, the writer could say because --- (and state the main reasons), or by --- (and express the actions mentioned) and then reword the thesis.  In addition, a "clincher" is used; this is a final thought that finishes by referring to the motivator.  Thus, the conclusion (a) reminds the reader of the thesis, the main point, of the essay and (b) it gives the essay a sense of finality.

A good essay will have coherence.   Coherence can be achieved by providing an explanation of the support, reminding the reader of the opinion in the topic sentence by repeating it or using synonymous words that suggest the opinion, and by using transitions. Unity is also required; that is, the ideas will make sense and they will follow in a logical manner. To achieve this logical order, each idea in the paragraphs should clearly support the topic sentence.  If ideas are irrelevant to this main point, they should be removed.

Academic writing demands that Standard English be used, no contractions be used, and correct spelling be used. Sentences will be varied in structure and complexity and transitional words or phrases will join them.  Vocabulary should also be appropriate to the scholastic level of the essay.

As stated in another post, most essays are written in an objective voice, using third person.  However, for a narrative essay, or one about a personal experience, first-person may be employed.  Avoidance of second person is encouraged. Unless directly instructing the reader, do not use "you." (or until you become famous as Hemingway and others have done)

In general, an essay should begin with an introductory paragraph.  This paragraph should start with some type of hook that draws the reader in and finish with a thesis statement explaining what the paper will prove.  The middle paragraphs should provide evidence and support for the thesis statement.  It is important for the evidence to be explored fully.  It is also important that there is sufficient evidence to support the thesis but not so much evidence that the reader is overwhelmed and the point is lost.  Each paragraph must further the point of the paper.  The final paragraph should be some sort of conclusion.  This paragraph should neatly wrap up the discussion and leave the reader with something to consider.

For any type of essay, you need support. For example, if you are writing an argumentative essay, you will be making claims about a certain topic. In order to back up your claims, you need support of some form. It can be research, an example of your own, or an important quote that relates in some way to the point you are making. If you are doing an analysis of a piece of literature, you need support from the text you are dicussing to prove any points that you are trying to make.

Another aspect to consider when organizing your essay is using transition words.  Transition words can indicate sequence, order, importance, or relationships between ideas you have used.  One good rule of thumb is to use a transition word or phrase when introducing new ideas or paragraphs.  You can also use transition words to help you summarize or conclude information in your essay.

In an academic essay, the introductory paragraph ought to clearly state the question the essay seeks to answer. Usually, the introduction will state a question (or a topical issue) and also state a proposed answer to the question.

For instance, an essay onAnimal Farmmight take betrayal as its topic. The question the essay will seek to answer will be something like, "Is every rise to power procured by betrayal?" The answer will be either yes or no, and the body of the essay will explore examples of characters coming into power and examine the role of betrayal in each example.

For me, the biggest rule is to have your essay well organized.  That is, you need to have each paragraph be about one thing and one thing only.  Many essays are without structure and seem to have random facts and ideas put together as the author thinks of them.  This is very ineffective in terms of making the essay easy to understand.

This is a very broad question and the rules could go on seemingly forever.  I'll try to just touch on a few of the big organization points.

Standard English essays usually start with an introduction which contains the thesisand main arguments or pointsof the essay.  The bodyof the essay follows and elaborates on each main point.  The conclusion of the essay is at the very end.  It sums up the thesis and main points and typically gives a final conclusive point to end the entire essay.

Standard English essays are written in an objective voice which attempts to avoid personal pronouns, opinions of the author, and emotional bias.