When you are asked to write essays for university classes, you are usually being asked to write arguments. An argument is a proposition plus some reasons why the writer thinks it true: "I think that X is the case, because A, which implies B, which leads to C," where A, B, and C are sub-topical propositions containing items of evidence that support the truth of X and that follow one another in a significant, developing order. Following are some basic elements of academic argument (whatever terms you use for them):

1. Thesis: your main insight or idea about a text or topic, and the main proposition (though it may have several parts) that your essay demonstrates. It should be true but arguable (not obviously or patently true, but one alternative among several), be limited enough in scope to be argued in a short composition and with available evidence, and get to the heart of the text or topic being analyzed (not be peripheral). It should be given early (not just be implied -- though its fullest and sharpest statement may be withheld for a time), and it should govern the whole essay (not disappear in places).
2. Motive: the reason, which you establish at the start of your essay, why a reader (someone besides your instructor) might want to read an essay on this topic, needs to hear your particular thesis argued and explained.* How is it that your thesis isn't just obvious, that other people hold or might hold other views (which you think need correction or adjustment) or need enlightening. Your motive won't necessarily be the reason you first got interested in the topic, or the personal motivation behind your engagement with it, which could be private and idiosyncratic: indeed it's what you say to suggest that your argument isn't idiosyncratic, but rather of interest to all serious students of your topic. Nor should the others you posit who hold another view, might be puzzled or surprised, might have missed something, be straw dummies. Your motive should be a genuine: a misapprehension or puzzle that an intelligent reader would plausibly have and argue for, a point that such a reader would really overlook. Defining motive should be the main business of your introductory paragraphs, where it is usually introduced by a form of the complicating word "But."
3. Evidence: the data -- facts, examples, or details -- that you refer to, quote, or summarize to support your thesis. There needs to be enough evidence to be persuasive; it needs to be the right kind of evidence to support the thesis (with no obvious pieces of evidence overlooked); it needs to be sufficiently concrete for the reader to trust it (e.g. in textual analysis, it often helps to find one or two key or representative passages to quote and focus on); and if summarized, it needs to be summarized accurately and fairly.
4. Analysis: the work of breaking down, interpreting, and commenting upon your data, of saying what can be inferred from the data such that it supports a thesis (is evidence for something). Analysis is what you do with data when you go beyond observing or summarizing it: you show how its parts contribute to a whole or how causes contribute to an effect; you draw out the significance or implication not apparent to a superfical view. Analysis is what makes the writer feel present, as a reasoning individual; so your essay should do more analyzing than it does summarizing or quoting.
5. Keyterms: the recurring terms or basic oppositions that your argument rests upon, usually literal but sometimes metaphors. These terms usually imply certain assumptions -- unstated beliefs about life, history, literature, reasoning, etc. that you don't argue for but simply assume to be true. An essay's keyterms should be clear in meaning (define if necessary) and appear throughout (not be abandoned half-way); they should be appropriate for the subject at hand (not unfair or too simple -- e.g. implying a false or constraining opposition); and they should not be inert clichés or abstractions (e.g. "the evils of society"). The attendant assumptions should bear logical inspection, and if arguable they should be explicitly acknowledged.
6. Structure: the sequence of main sections or sub-topics, and the turning points between them. Your sections should follow a logical order, and the links in that order should be apparent to the reader (see "stitching"). But it should also be a progressive order -- there should be a direction of development or complication, not simply be a list or restatements of the thesis ("Macbeth is ambitious: he's ambitious here; and he's ambitious here; and he's ambitious here too; thus, Macbeth is ambitious"). And the order should be supple enough to allow your to explore the topic, not just hammer home a thesis. (If the essay is complex or long, its structure may be briefly announced or hinted at after the thesis, in a road map or plan sentence.)
7. Stitching: words that tie together the parts of your argument, most commonly by (a) signaling transitions, acting as signposts to indicate how a new section, paragraph, or sentence follows from the one previous; but also by (b) by recollecting an idea or word or phrase used or quoted earlier. Repeating keyterms is especially helpful at points of transition from one section to another, to show how the new section fits in.
8. Sources: persons or documents -- referred to, summarized, or quoted -- that help you demonstrate the truth of your argument. They are typically sources of (a) factual information or data, (b) opinions or interpretation on your topic, (c) comparable versions of the thing you are discussing, or (d) applicable general concepts. Whether you are affirming or challenging your sources, they need to be efficiently integrated and fairly acknowledged by citation -- see Writing with Sources.
9. Reflecting: acts of pausing your demonstration to reflect on it, to raise or answer a question about it -- as when you (a) consider a counter-argument -- a possible objection, alternative, or problem that a skeptical or resistant reader might raise; (b) define your terms or assumptions (what do I mean by this term? or, what am I assuming here?); (c) handle a newly emergent concern (but if this is so, then how can X be?); (d) draw out an implication (so what? what might be the wider significance of the argument I have made? what might it lead to if I'm right? or, what does my argument about a single aspect of this suggest about the whole thing? or about the way people live and think?); (e) consider a possible explanation for the phenomenon that has been demonstrated (why might this be so? what might cause or have caused it?); and (f) offer a qualification or limitation to the case you have made (what you're not saying). The first of these reflections can come anywhere in an essay; the second usually comes early; the last four often come late (they're common moves of conclusion).
10. Orienting: giving bits of information, explanation, and summary that orient the reader who isn't expert in your subject, enabling such a reader to follow the argument. The orienting question is, what does my reader need here? And the answer can take many forms: necessary factual information about the text, author, or event (e.g., given in your introduction); a summary of a text or passage about to be analyzed; pieces of information along the way about passages, people, or events mentioned (including announcing or "set-up" phrases for quotations and sources -- see Writing with Sources). The challenge is to orient briefly and gracefully.
11. Stance: The implied relationship of you, the writer, to your readers and subject: how and where you implicitly position yourself as an analyst. Stance is defined by features such as style and tone (e.g. familiar or formal); the presence or absence of specialized language and knowledge; the amount of time spent orienting a general, non-expert reader; the use of scholarly conventions of form and style. Your stance should be established within the first few paragraphs of your essay, and it should remain consistent.
12. Style: the choices you make of words and sentence structure. Your style should be exact and clear (should bring out main idea and action of each sentence, not bury it) and plain without being flat (should be graceful and a little interesting, not stuffy).
13. Title: should both interest and inform. To inform -- i.e. inform a general reader who might be browsing in an essay collection of bibliography -- your title should give the subject and focus of the essay. To interest, your title might include a linguistic twist, paradox, sound pattern, or striking phrase taken from one of your sources (the aptness of which phrase your reader comes gradually to see). You can combine the interesting and informing functions in a single title or split them into title and subtitle. The interesting element shouldn't be too cute; the informing element shouldn't go so far as to state a thesis. Don't underline your own title, except where it contains the title of another text.

A footnote on motive:

*Why should your idea interest someone other than your instructor? Well, perhaps...

• the truth isn't what one would expect, or what it might first appear to be on first reading;
• there's an interesting wrinkle in the matter, a complexity;
• the standard opinion of the text, or a certain published view, needs challenging or qualifying;
• a simple or common or obvious-seeming approach to this has more implications, or explains more, than it may seem;
• an approach to this that may seem irrelevant, isn't;
• there's a contradiction or tension here;
• there's an ambiguity, something unclear, that could mean two or more things;
• this matter is difficult, or complicated, and needs some sorting out;
• there's a mystery or puzzle or question here that needs answering or explaining;
• we can learn about a larger phenomenon by studying this smaller one;
• published views of the matter conflict;
• this seemingly tangential or insignificant matter is actually important, or interesting. And so on.