What is a sole proprietorship?
A sole proprietorship is one of the four general ways you can choose to organize your business (the other three being as a partnership, as a corporation or as a limited liability company). Unlike the other three business forms, sole proprietorships are not a separate legal entity - they are indivisible from their owner and, as far as the law is concerned, the business and its owner are one in the same. As a result, sole proprietorships are the easiest and cheapest type of business to setup and operate. However, this also means that the owner is fully responsible for the company and there is no form of limited liability - so if the business gets sued or defaults on some debt, the owner is always on the hook for that liability.
If you are setting up a sole proprietorship, you must be the only owner (except that your spouse can be a co-owner in the company) - if your business is going to have more then one owner, your cheapest and easiest option is generally to start a partnership. As for running the sole proprietorship, you can choose to run it under your own name, or create a fictitious name to use for the business (and generally, in most states, you will have to make a fictitious name filing with the relevant state entity).
If you choose to start your business as a sole proprietorship, you can always change its form to a more complex form (like a partnership or an LLC) later, if you a need to do so arises.