I am sorry to say that this is a very large topic and there really isnt enough room here to explain it all. I have time to equip you with more questions, you can always email me if you need more information.
There are Proprietorships where you may only require a business certificate from the county.
There are Partnerships which are simple to form but you need to trust your partner with your entire life and that of your chilldren.
There are LLCs, and LLPs which are relatively new and very popular. These allow you to be protected from personal liability in business suits under most situations but not in all.
There are Corporations which can be S corp s which is designed to be more manageable for small businesses though like incorporations require regular filing of statements, and the keeping of minutes from board meeting, and a clear assignment of owenership percentages...