Can someone else use your business name?

Your question involves the law of intellectual property, particularly trademarks, which is much too complicated to cover in one email, but I'll try to give you a short overview.  You don't indicate that you have registered the name of your business as a trademark, which would provide you with some protection, and it is questionable whether you would be able to do so because the name (Munchies Pizza and Wings) may be deemed merely descriptive and thus not qualified to be a trademark.  For example, the name "McDonald's", as applied to fast food, is obviously fanciful, and (assuming other factors are met) can be registered as a trademark; the name "Best Buy", however, is merely descriptive, and can only receive trademark protection if the public associates it with one particular business.  Even then, Best Buy probably can't prevent one of its competitors from claiming to offer the "best buy in electronics".  Whether a name like yours qualifies as a trademark basically depends on whether it has acquired "secondary meaning", meaning that consumers associate the name with your particular business and no other, so that the other store's use of the name is likely to confuse consumers.  Not registering does not mean that you can't sue for trademark infringement, it is just that it will be harder for you to prove your case.  One problem you may have is that you apparently did nothing to protect your rights when the other pizza guy used the name last year, which may mean that you have waived any rights you might have.  Another problem is that litigating a trademark case, particularly one where you have to prove secondary meaning, can be very expensive.