The answer is, of course, yes. Many young writers have been published, including Christopher Paolini, who started writing Eragon at age 14 (he published at 19).
Some publishers prefer not to deal with younger writers because of the additional work at the contract stage–writers under 18 cannot sign their own contracts. There is additional work to work out the how the rights to the work are assigned, and how the payments will be handled (typically to a trust until the writer turns 18).
However, there’s a good marketing angle for a young writer: if a publisher prints a manuscript written by a young writer, people pay attention. People are attracted to what’s new, and a new book by a new author still in school gets attention.
Also, you’re not required to (and I recommend you do not) reveal your age until a publisher accepts your manuscript and you move into the contract stage. There are legal considerations that must be made when the author is too young to legally sign the contract.
As with all authors, as a young writer, you want to consider hiring an agent to help you through the publishing process. They can help you understand all of the legal mumbo-jumbo associated with the publishing process, for a percentage of any money you earn (usually 10-20%). Agents will also know who is more likely to accept a submission from a young writer.
As an aside: Don’t ever pay an agent up front. All legitimate agents work on a 10-20% commission–they only get paid when your novel sells. Their job is to sell your novel, so they should only get paid when that’s successful.
Make sure that you have your parents involved as well–as a young writer, people will be looking to take advantage of your inexperience. (By the way, inexperience isn’t bad. Everyone is new at first.) Your parents (and an agent) can help you avoid many of the pitfalls that plague new writers.
Before you submit your manuscript to either an agent or a publisher, you’ll want to make sure it’s really ready to publish. This means that your work needs to be edited for inconsistencies, errors in grammar or punctuation, and overall style.
Don’t trust just yourself to find all inconsistencies and grammar/punctuation errors. Use friends, family, or (if your parents are willing to spend some money), use an editing service. I often recommend that aspiring authors take advantage of at least a sample edit in order to determine how close their work is to being ready.
Check back next week for details on the overall process of sending work in to publishers.
Best of luck as you continue your work. Write on!