A private company is any company that is not publicly held, meaning that it could be a corporation, a partnership, a LLC or a sole proprietorship.
Forming a sole proprietorship is easy. Repeat after me: “ I now have a company and it is named ____ (insert name)” You now are a sole proprietor. You may have to register your “fictitious” business name, you may have to obtain a business license from your City/County/State, and you may have to apply for various registrations, such as sales tax, property tax, payroll tax, etc. But you are in business.
Forming a partnership is almost as easy. You and others agree to be in partnership. You are there. Of course, only a fool would do that without some sort of written partnership agreement, since people tend to have short memories about what they agreed to, and partnerships tend to end up on the rocks as the result of unfulfilled expectations and misinterpretations. Partnerships are dangerous because they expose you to the liability created by your partners, amongst other things, and if you are determined to have one, consult with an attorney, please. We don’t recommend them. After you have formed a partnership, you’ll have to go through all the registrations mentioned under sole proprietorships.
Forming a LLC requires a tad more effort. You will have to register with the state you are in, and then follow the same business registrations for a sole proprietor. You should work up an “operating agreement” especially if you have other owners. Consultation with a CPA and attorney is recommended.
Forming a corporation requires application to the state for a charter, and then various forms of paperwork including minutes of directors meetings, shareholders meetings, bylaws, etc. There are lots of formalities which must be adhered to, and failure to do so will invalidate the corporation as an entity. Consultation with a CPA and attorney is virtually mandatory. Doing an incorporation on-line by yourself is similar to doing your own appendectomy, maybe possible, but not recommended.