You usually present your answer to the prompt in a thesis statement. A thesis is a statement that explains your argument specifically. If you have a strong thesis, it is much easier to write a strong essay. You support your thesis carefully with evidence.
akannan has succinctly stated the best method for answering an essay question. Essentially, your response is a persuasive essay, for you wish to convince the instructor that there is reason to believe your response: your elaborations substantiate what you have written in your answer/thesis statement.
For instance, if you are asked to define Oedipus Rex as a tragic character, your response/thesis could be strong and to the point: In the play by Sophocles, Oedipus Rex is a tragic character because he adheres to the classic definition of a tragic hero as defined by Aristotle in his Poetics. That is, Oedipus is a man of high stature, he possesses hubris, and he commits a tragic mistake (hamartia).
[After you have made your thesis/answer, support each point with details and citations from the play:]
That Oedipus is of high stature is apparent from the play's beginning, for he has solved the riddle of the sphinx and has been made King of Thebes. ....
There are a few things that you can do to present your answer.
First, you should identify the question that you are seeking to answer. The better the question is formulated the better the answer will be.
Second, you need to set parameters. Tell the reader what you will be addressing and what you will not be addressing.
Third, you will need to formulate a thesis statement that addresses the first two points.
Finally, you will need excellent reasons of why you believe you thesis is a good and true one. This is where you seek to convince your reader what you "really" think.
In presenting your answer to a question which is in an essay format, I would identify a couple of points that have to be made. The first would be to ensure that you are fully answering all parts of the question. Make sure you have completely addressed the components of the prompt and this will ensure that you are "really" answering the question. The second thing I would do is ensure that you are able to substantiate your answer with evidence and analysis. The essence of the essay task is to ensure that sufficient detail and support has been given in terms of writing a formal and composed response to the question at hand. This would involve being able to synthesize information from outside sources and asserting your own analysis with substantiation and thought in the process. I think taking these steps would significantly help in ensuring that you are "really" answering the question.