If I trademark a business name that has not been trademarked, but has business’s in other areas and states with similar names, what happens to those other business’s?

If the business is trademarked, then you own that mark (that specific mark).  If businesses in other areas or areas near you (it makes no difference) share a similar business model or the products can be confused with each other, it may come down to who was first and whether you are encroaching on any of their rights. I would consult with an intellectual property attorney before you begin this process to not only protect yourself but to ensure your business won't be inadvertently taken by other businesses because people will confuse your trademark with existing businesses.

If there are businesses with similar names in similar lines of work, you may not be able to trademark the name--even if the name has not been registered previously, if there are already people using it in commerce, you generally cannot trademark it. (And if you did and tried to take action against pre-existing businesses using the name, they may be able to invalidate your mark on grounds that it should not have been given due to their prior use.) Also, it's often the case that names have not been trademarked because they are too generic or descriptive to be trademarked, like, say, "Arrow Archery Supplies."

You should come up with a name that has not been used by others in your line of work, then work with an intellectual property attorney to make sure it is trademarkable, if you are concerned about protecting the name. If the name is not really an issue--think about how many "Brothers Pizza" places there are out there; the value for those business is in the business itself, not the name--then don't worry about it and just run your business.