I have final exams coming up and one is an essay test. It is two questions, but our teacher didn’t tell us what the questions were.
Please give me your tips. 
When you get the question, stop for a moment and think about it before you start writing. Figure out what the angle is-- an essay question always has some major point behind it that hopefully you should be able to pin down. It might be compare and contrast or something else, like outlining the main points of an issue that was described in class. Think about how those main points might fit together in a logical or at least tidy manner-- this will be the structure. Might only be three main points-- so three sort of paragraphs. Once you’ve worked this out, write the first sentence or two with the basic thrust of that big issue to communicate to the reader that you have a damn clue. Then start writing on those main points you thought about, in order, from one to the next. Every couple of sentences or so stop and look back at the question and consider whether what you’re writing has anything to do about that question-- don’t go stream-of-consciousness.
Think of how much time you have. Part of the challenge is to get all the main points in an appropriate depth to the time allotted. It’s a timed exam, I assume, so make sure you say what you NEED to say, then spend extra time with the extraneous stuff.
From what I hear, pretending the teacher knows next to nothing about the subject matter, then going to explain in detail helps, especially if it’s an essay made to show the teacher what you know.
Writing an in-class essay is a lot different than writing a take-home essay.
The first thing is to plan your essay. Get a basic idea of what points you’d like to make and what evidence you have to support those points. Then think of how you are going to introduce your argument and conclude it. Usually you can rework the essay question into a reasonable thesis statement.
That said, a good key is “Tell them what your going to tell them, then tell them, then tell them what you told them.” This holds true for the essay on a whole as well as individual paragraphs. Use the opening paragraph/sentence to introduce a point, and then make the point using as much evidence as you can, and then write a conclusion that places the point in the broader context of things.
Sit down before the essay and memorize the writers and titles of all the texts you used in class. You ought to be able to cite things somewhat properly. Ideally, you should also memorize a few broadly applicable quotes. I’m assuming you are in college and in a college level in-class essay teachers are going to look at how much you understand the course material and how much evidence you can come up with for your points. The more you can bring in the source material, the better.
A good tip for all essays is to include in your closing paragraph the limitations of your essay (“My analysis of the impact of hot dog consumption doesn’t include cheese filled hot dogs…”) and/or suggest avenues for new research.
Also, learn how to use transition words. These are the raw materials that make an essay work.
Good advice above.
Remember your fundamentals: introduction (thesis statement), body, conclusion (repeating thesis statement in different words).
Some essays (particularly for language classes) are graded just as harshly for grammar and punctuation as content, so be wary of your spelling and verbiage snarls, etc.
Seconding the advice above to write it as if you were writing to someone who had no prior knowledge of the topic. Always a good strategy.
Some good advice above. However, if you are really low on inspiration and it’s an arts topic you can use my “trick”.
You find one critic who holds a very extreme view and another critic who’s opinion is the complete opposite of the first critic’s. You outline both lines of thought and conclude that truth was somewhere in the middle. This approach has always gotten me a “B” and the comment that it was a very balanced essay. Of course you’ll never get an “A” using this approach but it can work if you don’t have any strong feelings on a topic and don’t have the time or the will to develop one.
Of course you won’t actually know what your questions will be, but it still might be a good idea to head to the library and look for some of the more extreme views that are around in your subject field so you can sprinkle them in to make whatever you are saying look levelheaded and sane. 
Good advice above, but it can never hurt to have more. I had 10 cups of coffee while working with my writing partner tonight and I am not going to bed for hours, so here is my take.
The important things that helped me get through my AP English exams in high school (and I got 5’s even though I can’t spell at all and have a liberal attitude toward punctuation) were knowing how to create a strong thesis and having a good handle on essay structure, . If you know what you are going to say and how you should say it, the rest is just filling in the blanks. (I am going to bias this a bit toward argumentative lit essays as those were what I wrote the most of in college and are what I remember best, but the basics apply everywhere.)
The very first thing you want to do is figure out your thesis. Its ok if this changes a little during the essay writing process, but you have to pick a point of view. Read the question, figure out what it is asking and then figure out what you think about the answer. By way of example. I had an essay to write in high school about Oedipus Rex. The question we were given was something along the lines of. What was Oedipus flaw that made him a tragic hero. I was a bit of a prick in high school so my thesis was “Oedipus’s tragic flaw was that he was a jerk.” Granted this is not a position that won me any points with my teacher, but it was supportable and (importantly) suitably narrow. Most people tend to pick thesis statements that are far to broad to effectively answer in the course of an essay. Keep your thesis to a position you can both argue and support in the time frame you have. Don’t get too hung up on the thesis though. Remember all a thesis has to be is your opinion. If you have an opinion on a topic you are 90% of the way there. Now that you have your thesis take a minute and think of three reasons why you believe your thesis to be true. These are your points. Write them down, we will come back to them.
Once you have your thesis you are ready to write your intro. (Hint: in many high school essay exams you can create a thesis by restating the question. It won’t work forever but it will make your life easier now)
Your introduction usually doesn’t have to be much more than a vehicle to present your thesis (I am assuming you are in high school), plus a brief summary of what will follow in your essay’s body. Where you place the thesis is kind of up to you. Placing it in the beginning is easier, but stylistically I prefer it at the end of the intro. Don’t place it in the middle, teachers hate that. Also make sure that somewhere in there you have a sentence that lists the three reason (or more) that you believe your thesis to be true.
So you have the intro and thesis. On to the body.
What should be contained in the body paragraphs? First off you should start each paragraph off with a point you would like to make about the thesis (the three reasons mentioned above.) These are referred to as topic sentences. You want to make points that support your thesis. For any good essay you should have at lest three points you are going to make, more is better if you have the time, but three is the minimum. For the thesis above “Oedipus is a jerk” I would say make the point that we know he is a jerk because,
A) he has a bad temper
B) he doesn’t listen to good advice
C) he doesn’t trust anyone
Each one of those letters will become a paragraph. For each point you also need proof, generally a supporting quote or passage from the text to show you aren’t just making things up. Again you should have at least three proofs (or more) for each point. (but honestly you can probably get away with only 2 for some of them).
The next bit may seem obvious but it will truly make your life easy. It reduces essay writing to connect the dots. Just fill in the blanks and the rest is simply style. In case you are just skimming:
THE FOLLOWING IS THE MOST IMPORTANT THING I WILL SAY!!!
The key to writing the body paragraphs is following this little formula (which I was forced to repeat on a daily basis by my senior AP teacher).
Point-Proofs-How Proofs prove point- How point connects to thesis,
Rinse and repeat.
Remember this formula, and make sure you don’t leave out a step. You would be surprised how many essays I have seen that make a point, give proof and then don’t tell you why you should give a damn. It is not enough to say that Oedipus has a bad temper because he killed a man for not getting out of his way. You have to then say why this proves he has a bad temper, and what having a bad temper has to do with being a jerk. Again, if you know your topic and have an opinion this won’t be hard. Just fill in the blanks.
Take topic A, give proof that it is true. Say why proof proves it to be true. Give second proof. Say why second proof proves it to be true. Give third proof etc. Then when you are done with proof say why the point is relevant. Move on to the next point, do the same thing. You can crank out a body in no time flat using this method.
Lastly conclusions. I like even sven’s suggestion of saying what your essay left out. I also would add that you should say why the reader should care about the thesis in the first place. Conclusions are tricky because you want to put all the pieces back together again without restating the thesis or summarizing the essay. I tend to take this opportunity to say things about the big picture. Things like, “the lessons leaned by Oedipus are ones from which society at large could benefit. In life it is always good to remember, don’t be a jerk” Or something schmaltzy like that. Teachers eat that stuff up with a spoon. It proves you are thinking about more than just what they asked you about.
A word on pre writing organization:
I tend not to work from an outline but writing one may help you stay on track and work faster. I know it helps a lot of folks, and it can be a good way to get a handle on what you are thinking. Formulate your outline as a stripped down version of your essay in “formula” form.
I.Thesis
A) point
1)proof (a little note to yourself that you can expand upon in the essay)
B)point
1)proof (and remember there should be three of these, three points too)
etc
Did you actually read all of this?? If so I have one last thought. I don’t think it is a good idea to assume that the reader doesn’t know the subject. I think the better idea is to assume that the reader is familiar with the subject, but really doesn’t understand it at all. It is a small difference, but an important one. If you start thinking the reader doesn’t know the material you will start to summarize instead of argue your points, and that is very bad. So stay on topic, follow the formula and you will be fine. The best way to study is just to know as much about the material as you can so you will be able to have an opinion you can argue.
Finally save out a few minutes at the end to reread your whole essay carefully. This is especially important with in class essays. When you are writing quickly it is easy to make simple mistakes like leaving words out of sentences. Proof read and correct that stuff.
So there you have it 2 years of AP English encapsulated into a single post. Hope you get some use out of it. 
For me, the introduction is always the hardest. Once I get the introduction nailed, everything flows. The introduction is like an abstract, a summary of your entire essay. Usually, I use it to state the premise of the essay and a quick, one-sentence (or slightly more) summary of my points.
Next you try to gather content for your essay. I am not sure what type of essay you are up against, but if it is an arugmentative one (“Is a better than b? Discuss” Or “Has the advent of a influence b? Agree or disagree?”) then pick a side – defintely the side which you have more ideas for. Be sure to explore both sides of the issues. And be sure to be consistent with all your points. If possible, use examples to illustrate your point.
It never hurt to scribble down your thoughts, be it in points-form, mind-maps or some choice kewl sentences which you have thought off. Always spend some time planning your direction and approach.
If all goes well, you shalln’t have any problem with the conclusion. Good luck!
Write a strong hook and thesis and your entire paper be set up. Someone already commented with a good format for a thesis.
Write three paragraphs using the points of your thesis, and use quote evidence to support your points. Introduce the quote, then say the quote, then analyze the quote and why it supports your idea. Look up “quote sandwich” for more detail!
Conclusion either wraps up what you said, or introduces a new idea. Always restate your thesis.
Welcome to the Straight Dope. Presumably the original poster, who has been banned, finished the essay 13 1/2 years ago when this thread was opened.