The word "essay" comes from the French, and it originally meant something like "an attempt." An essay is an attempt to express ideas or feelings about one thing or another. It started with Michel de Montaigne (1533-1592) and became popular with writers and readers because the form offers so much variety and so much freedom of expression. There can be no hard and fast rules for writing essays because they are supposed to offer freedom for the writer to say whatever he or she feels like saying--or trying to say.
What I am writing now is a sort of essay. I am trying to say put some thoughts and feelings into words without knowing exactly where I'm going--and not particularly caring, either. In writing an essay you discover what you think while you're writing. You can't think the whole thing out in your head and then write it all down. You have to think and write, and write and think.
Emerson wrote a lot of essays, and he says somewhere that the writer doesn't always know where he is going and often ends up someplace other than where he expected to be. So essay writing is a sort of adventure, a voyage of discovery. You can make up your own form and have some fun. There should be an introduction and a conclusion, but you don't have to write the introduction first and the conclusion last. A lot of people get stuck with introductions--because how can they know what they're going to say before they've said it?
You need to get words on paper. You can always rearrange them, and it's really easy if you're using a word processor. Read the last chapter in Strunk and White's The Elements of Style. It's a very short book, and it's full of good advice.
James Thurber was one of the best prose writers in America. He has often been quoted as saying:
Don't get it right. Get it written.
Thurber confessed that the first drafts of his essays and other short pieces looked like they had been written by somebody who was drunk on cooking sherry. But he was a perfectionist, like his friend and colleague E. B. White. Thurber honed and polished his words until he was satisfied with the final draft. No doubt he wrote many drafts before that happened. A writer should expect to write several drafts of any piece of writing.
Holden Caulfield has some good advice about writing essays for school assignments in The Catcher in the Rye.