A mouth piece is a pretty essential and basic piece of equipment for a boxer. In fact, the mouth piece was actually first invented for boxing. Originally called a "gum shield" (and often still called that in the boxing world), this amazingly important piece of boxing equipment was invented in 1890 by Woolf Krause, but didn't really become popular in America until the 1950's and 60's. It's kind of insane for me to think of boxing without this, but boxing's been around for pretty much as long as people have and it was very, very recently invented! (This is not to give you any excuses to ever skip wearing one! It was just given as interesting facts!)
Since the invention of mouth pieces, though, they've grown to be used in practically every sport with even a little bit of contact. This means there are different kinds. For a long time I wore a double (an upper and lower attached together with a hole to breathe out of in the middle). This was stupid and I wish I would have been made to stop. It's hard enough to breathe in the thick of it all without cutting off most of your airway. I now wear a cheap Everlast single, bought at almost any store for $5. This is not the piece I'd recommend for your average boxer, but I have a tendency to lose mine, leave it in heat so intense the plastic gets wrong, or chew through it when I'm grinding my teeth hard..
There are a million (seriously) options if you type in "single mouth guard boxing" on Amazon of fun and crazy versions of these at home boil and fit mouth guards. I honestly just haven't upgraded because I don't get one until it's lost, and don't have time to order a new one online in between. Once I buy a cheap replacement (telling myself it's just until I order), I get busy and don't order. Don't be like me. Invest a little time and energy into getting the right mouth piece. I hear it's amazing, and I know it's incredibly important. The day I lose a tooth or get my jaw broken I'll be really mad I hadn't taken five minutes before then to do this.
So with that advice of taking the tiny, tiny bit of extra time and money, I refer you to your dentist. Most dentist offices will create mouth pieces from a mold that they take of your mouth. If you are like 80 percent of boxers who don't have health insurance to take on the cost of a dental visit, there are a lot of online companies where they send you a kit to take your own mold, send it back to them, and then they send you a perfectly fit mouth piece. Especially if you're talking about a child boxer, DO THIS. Do not risk using a cheap mouth piece. It's not just about how much protection they offer, it's also the fact that these won't get knocked out and won't restrict your breathing or talking. One company I can recommend is The Mouth Piece Guy (click the link to be directed to their website and get started).
The last advice I can give is to make sure not to purchase a red (or pink, or close to red) mouth piece. They are illegal to box in, and the last thing you want is to have taken the time to get a properly fitting mouth piece and find yourself headed to Wal-Mart to purchase and wear an un-molded one during your first match.