Before you start writing, make sure to read the prompt carefully. It is very important that your essay correspond to the actual requirements of the school. Some honors college essays may have word count limits, and each school is likely to have its own requirements. If you are applying to more than one school, be sure to tailor your essay to each college's requirements. Submitting an essay that clearly ignores the school's guidelines will reflect poorly on you as an applicant.
After you feel you understand the prompt, take time to gather your thoughts and figure out what you want to write about. Make sure that your chosen topic or anecdote clearly explains a positive aspect of yourself. Remember, the point of the essay is to sell yourself to the admissions committee.
If the prompt is something like "Why have you decided to apply to the Honors college?", take the time to think through your real reasons. While being honest is very important, it is also imperative to make your reasons engaging and to present yourself in a positive light. If you just want to go to the honors school because the students get nicer dorms, don't say so! Talk about the ways being in the honor school will help you achieve your life goals and advance yourself as a person.
Many people find it helpful to plan out their essay before starting to write. This can help you to organize your thoughts into a clear, coherent narrative. Making an outline is an excellent way to do this, though there are other methods. Whatever you do, make sure you understand where you're going with the essay before you start writing it.
Your first draft might not be great. It might be terrible. But that's alright, because you have plenty of time to revise. It's much easier to improve an essay on paper than one that's still floating around your head.