I need advice on this rather desperately — if someone would be so kind as to give some.

Pretend this is an essay. You are the teacher, and I have just handed a sheet of paper to you.

And this is how it reads:

Introduction: When I come into the world, I am a screaming, squalling baby. Born two weeks after the due date, I am about as not-frail as a baby can get. I live with my parents in an apartment overlooking the city until I am three — and then we move.

¶1: I move a lot, after that — never staying in one place for more than three years, making friends and then disappearing off their radar. I'm only in still contact with one, a friend from second grade, and even that connection is fading. I don't think about things like love, boyfriends, marriage — that's not even on my radar yet, even though other children my age are starting to think about that sort of stuff.

¶2: At ten years old, my family moved to a place in the North West — we decide that we'll probably stay there. After a year of homeschooling, we begin to test the waters of alternative schools, and, two years in, find one we like. I begin to make friends, and I realize how much I've been missing. I also discover my love of science.

¶3: Now I am almost fifteen, and I have been doing quite a bit of soul-searching. For the first time that I can remember, I have friends who care about me and vice versa. I have decided that no matter what I do, it has to be outstanding — I have to be better, to live a greater life that changes the world. And I realize that I might be having a crisis in regards to sexual orientation. Now, I realize pretty quickly that I'm not attracted to boys, as I once thought, and, as a girl myself, I can proudly consider myself "not-straight."

Conclusion: Now comes the hard part — when do I tell my mother? My family is supportive — we have friends who are all over the LGBT spectrum, and my mother's best friend is married to another woman. But I'm a coward — I don't know what to say, or when to say, and whenever given an opportunity, I almost say something, but I keep my mouth shut. I was originally planning to not say anything, and just bring home my partner for my mother to meet, but I'm realizing I can't really do that, especially since she keeps making joking jabs about a guy-friend who I go to the movies with. I have given really no indication that I might be anything but straight, so I don't really blame her — but — what do I do?

Chill, don't over think this, don't make such a big deal about it, I'm betting 90% chance your family already knows that you're homosexual but are waiting for you to be ready to tell them and act as if they don't know.
It was exactly the same for my sister.
Just say it in passing, maybe before going out say I'm going with my girlfriend or something. No need to make a big fuss about it if you dislike being awkward, it's your family they won't stop caring for you because of a simple reason like that.

Whether you say it in passing or come out in a letter is your decision. I came out with a 13 page long essay that my parents didn't read, so it's really great that you're keeping it brief. They won't read it if it's too long.

I think it's good, but from a grammatical perspective, I think you need fewer dashes. Semicolons would work wonderfully where you've placed many of your dashes (if you don't know how to use a semicolon yet, Google it! I promise they're not that scary!).

You talk a lot about your childhood, but not very much about your sexual orientation at all. While your childhood can help to explain how you discovered yourself, it's taking up a lot of the essay.

Since you talked about your childhood, why not tie in your sexual orientation? Talk about how, at age 3 when you moved to a new home, you were torn away from a cute girl. Or, if you had an LGBT celebrity you looked up to during tough times, you could mention that, too.

I hope I don't sound like I'm attacking you. These are just ideas you can take or leave as you see fit.

If I actually had understanding parents, I'd be in the exact same position as you. Even the cowardly and timid part. My best bet? What would I do? Make some cookies. Then write the word 'GAY' on a cookie and make sure she sees. Then I would say 'You are what you eat' and eat it. If that's too much; I'd take a picture and print it out before writing a caption on the bottom, and sticking it on the fridge

Thanks to everyone for your advice! I was actually going to do what Aries said, but I ended up coming out before that due to a family crisis.