If there's a form, use up most of the space allotted. On my son's forms to prep schools, it's about a paragraph. (Not enough, imho!) However, if you are supposed to do it on your own paper, call the admissions director and ask. It's really not a bad thing to call and ask! Or if your counselor at school knows about these things (all private school counselors would know), ask them.
Oh, and if the school has competitive admissions, and you have good handwriting, you might want to type up your essay and then handwrite it so they know you did it. (At the most competitive schools in my area, they have short essays handwritten only because they know parents were likely writing them. That's LA.)
If you have to do an essay with the entrance exam (ISEE, HSPT, etc) that they send to the schools, you should try to do one of those silly 5 paragraph essays with the transition sentences at the end of the paragraph and at the beginning of the next paragraph, ETC if that's what your English teacher has taught you. (Oy, who can read those things, drives me crazy and writing is my profession!)
If you can't get an essay done in the time (30 minutes for my son's exams), then it's best to write a good shorter essay with a conclusion paragraph and not just leave the essay unfinished. Practice!
What have you achieved with your writing skills? Think of something specific and see if a thesis pops in your head.
doing the private high school thing right now