As others have noted, no, you do not have to create a business of any sort in order to write and publish.
Copyrights are usually registered in the name of an individual, not a company. Look at the copyright notice in the books on your shelf. Almost all will be a person's name, not a business.
As a self-published author, the easiest way to get your books out there is to sell through on-line sites like Amazon and Barnes & Noble. They then take care of all the "businessy" stuff. As others have said, you'll get a 1099 reporting your income. As a copy of this goes to the IRS, it is a really really good idea to report it on your income taxes. To the government, the only crime worse than torturing and murdering innocent children is failing to pay your taxes.
The one thing that is going to be an issue is if you intend to personally sell your books retail, that is, sell books directly to customers rather than selling through Amazon or bookstores. In that case, you have to collect sales tax, which means you have to register with the state to collect the sales tax and forward the money to the state.
Personally, I set up a business to handle the sales of my books. It's just a sole proprietorship, not a corporation or an LLC or anything complicated. I had to file a form with the county and pay a registration fee, I think it's $100 for five years or something like that. All that really does is give me the right to use the company name rather than my own name on "official" documents.