What a teacher looks for will vary based on the teacher, the level, and the subject.  However, all teachers really want an easy to read, well-developed argument.  We almost always look for a thesis statement.  Proper grammar is also good.

Substance!  Along with the organizational skills mentioned so well by the above editors, teachers are absolutely weary of plodding through pedestrian thoughts that lead to stale and repetitive ideas and structure.  But, when a student thinks down a new avenue and supports his/her ideas with logical reasons or examples or facts, expressing her/himself in a fresh way with depth, and imagery and other figurative devices, voila!  There is an electric charge sent to that teacher and the substance light flashes in her mind.  "Now, this student has said something to grasp!"

Write the kind of essay that a teacher will want to read aloud to the class: fresh ideas that are supported in a clear, but original manner that employs figures of speech and innovative ways of expression.

All of the above posts give you good points to consider to write a good essay.  Post three is especially good in that the format of an essay is clearly explained and easy to follow.  The one thing I would add to all of the above is because I taught 8th grade, I think.  So often, students would forget about using the writing techniques we had practiced such as complex sentences, varied beginnings, reducing the number of one and two letter words, using show me examples rather than tell me, and combining low quality sentences into one good sentence.  This revision should be done after proofreading because then you will have to proofread again, catching other errors you missed.  Essay writing is complex, but gives the writer the chance to truly express themselves in their own writing.

A good essay is organized in a logical way:

• You tell the reader what you're writing about in your introduction (without saying, "I'm writing about . . .)
• You support your topic with good evidence in the body paragraphs.
• You conclude without simply repeating your introduction.

One very frequent mistake students make is failing to proofread. Essays that are filled with careless mistakes reflect badly on the writer. A simply proofread can catch most of the mistakes and save you a lot of points.

You also want to make your essay look good. Handing in a mess is going to hurt you. Take some pride in how your work looks to others.

While I agree that each assignment and each type of essay has it's own criteria, there are certain standards.  Of course, teachers will check to make sure that proper grammar and writing skills are used.  An appropriate and mature vocabulary are also necessary.  Most essays follow the standard format of introduction, body, and conclusion.  A thesis statement should be included in the introduction. 

Possibly one of the most important aspects of a good essay is the depth of the topic itself.  If the topic is too narrow, the writer will not have enough to say and cannot properly expound upon the topic.  They will likely just end up repeating themselves instead of properly proving the point.  If the topic is too broad, the writer is equally unlikely to prove their point.  A broad topic does not allow the writer to get down to the nitty gritty.  They will spend half the essay explaining the big picture instead of providing evidence to support their thesis.  While looking at the big picture can be helpful, it cannot take up the entire essay nor can it replace a specific point.

There are quite a few criteria I look for when grading an essay, and like the poster above stated, it depends on the purpose of the essay; whether it's to inform, persuade, entertain, etc.

In any type of essay, it's important to have a focus. Each paragraph should center around the same general idea and avoid meandering to off topic subjects. If a student is writing a narrative about an important time in his or her life, the essay should focus only on a short period of time rather than covering a large spand of time. This helps to keep the essay clear and focused. 

Complete sentences and proper grammar, spelling, and puncuation are always important. When grading, I tend not to take off for every single error, but if a paper is riddled with errors, it becomes distracting or even confusing to read and will inevitably cause the essay to lose points.

Each type of essay has its own set of standards.  The purpose of the essay determines what will be expected from the writer.

If the essay is informative, the writer will be explaining or teaching the reader about a subject.

to inform=Mediation yoga techniques have certain requirments to be useful in relieving stress.

The persuasive essay will be argumentative in nature because the writer will be taking a side on an issue.

to persuade=The United States is the greatest place on earth.

The literary analysis will take a literary work and like the persuasive essay and through research argue its value or lack of it.

to analyze=Death of a Salesman by Arthur Miller represents the best of all American drama.

First, when grading an essay, the instructor will expect all grammatical and mechanical errors to have been corrected. These are basic in writing and should be revised through reviewing of the work.

Secondly, the thesis must be clear and well-written.  In beginning, writers, the thesis should be found usually at the end of the introduction.

The next aspect of the evaluation process is to check that the boundaries of the assignment have been followed: length, type of assignment, and formatting.

Then, each part of the essay should be reviewed.

The introduction= Approximately 10% of the paper; includes the attention getter; introduction of the topic; the relevance of the topic; transition into the thesis statement

The body=The heart of the paper; in a beginning essay, 4-7 paragraphs; includes supporting information or evidence; properly documented (APA or MLA)

The paragraphs= Covers only one subject; connected using a transition; topic sentence; and concluding sentence

The conclusion=Summary of what has been said; transitional sentence; the conclusion statements usually leaving the audience with something to think about.

Then, the instructor needs to consider if the essay was interesting and well researched.  The totality of the paper should be considered.  Usually if the paper is especially important to the grade of the student, the paper deserves a second reading.  It takes time to give an essay a fair grade. 

One of the most important things to consider when writing an essay is that your writing addresses the topic or subject selected by the teacher.  Your introduction should clearly state your purpose and focus.  Too many times I have read student papers that were cleverly written, but had very little to do with the original prompt!