Get feedback on your piece. This may can be from a trusted friend, experienced writer, teacher, or writer group. Be open to their critiques and make changes accordingly. However, use your instinct when it comes to any major changes. If you disagree with an critiques, rewrite what you feel is appropriate.
Edit. Have someone who knows grammar and good writing edit your piece to be sure it's ready for submission. It is very important that your piece be error free. The content could be excellent but a publisher might throw it out for the poor grammar and spelling.
Do some research. Publishers have guidelines in place for how they want the piece to be submitted. Follow their directions; if you deviate they can toss your submission aside and move onto the next. Now is not the time to separate yourself from the crowd- let the writing do that.
Choose a publisher you feel is best for your piece. Do not send a non-fiction piece to a publisher looking for fiction.
Look into publishers that specialize in teen writing. Teen Ink, Teens@Random, and Blue Jean Publishing are publishing companies that work specifically with teen writers.
Pay attention to criticism. If your piece keeps getting rejected, consider what the editors have to say. Some will give you feedback, some will not. If you are fortunate to get a response, use their feedback constructively.
Self-publish the piece if you can't find a publisher. Lulu, CreateSpace by Amazon, The Overlook Express are popular self-publishing options. Self-publishing requires a bit more leg work- marketing and expense, but can be worth it. Publishing the piece solely as an eBook is a popular option; Smashwords and Scribd have easy upload systems to get your book in electronic format.
Enter contests for teen writers. Pomegranate Words composed a list of writing contests specifically for teens. Alliance for Young Artists and Writers and the Writing Conference Inc., for example, have yearly published contests for teens.
Go local. If you are having difficulties getting published on a larger scale, narrow your scope. Try your local newspaper or possibly a magazine that is specific to your area.