Finding the right name for your band can be an arduous process, and when you finally have that fist-in-the-air moment, you want to protect your new name with everything you have. It's not uncommon for people to wonder if they can copyright or trademark their band name to prevent other people from using it.
First thing's first - it's not possible to copyright a band name. You can only copyright the material you write. It is, however, possible to trademark your band name, but you have to register the trademark in every territory that you want to defend it in. If you want to stop anybody anywhere in the world from using your band name, you would have to register it as a trademark globally. This would be an incredibly (!) expensive and time-consuming task, and overall, not something that is worth the time or effort.
Think about it; if you build yourself up to the point where your band has an established name, and somebody else wants to use that name, are you really going to want to spend lots of time and money going through litigation to prevent them from using it? If somebody breaches your trademark, are you going to be in a position where you have the time and money available to hire a lawyer and go through a lawsuit? The process would be very expensive and time-consuming - all because somebody is using your band name.
Instead, you should be taking ownership of your band name and establishing it as well as you can. Get online and build a global fanbase so that whenever somebody searches your band name, you'll be the only thing comes up! Anybody else who is using the same name as you should be buried beneath your success. Do you think there are bands out there getting away with trying to be called "Iron Maiden"?
To summarise - technically you can trademark your band name in global territories, but it's much better to spend the time and energy building global awareness of the band and band name so that nobody else wants to use your name anyway.