Most single driver businesses are set up as Sole Proprietorships which means that you report your earnings from driving contracts on your personal tax return
Sole proprietorships are easy to set up and inexpensive. You can create a DBA (doing business as name) for your business such as Super Fast Courier Service and you should check with your local city or town to see how to register your business name.
The disadvantages of a sole proprietorship is that the driver / owner is personal liable for all business bad debts and judgements (if any) from lawsuits; If you have assets you want to protect, such as a home, and you are sued as a sole proprietor, any assets you own can be at risk. If you want to separate your personal assets from your business activities then look into creating an LLC (limited liability company). Some states do have extra taxes for LLCs so you need to check that out first when deciding how to legally set up your driving business: Sole Proprietorship vs. LLC? Pros and Cons
https://www.thebalancesmb.com/which-is-the-best-state-to-form-an-llc-4178032 (link to article on Best States to Form LLC in)