A partnership is a business relationship entered into by a formal agreement between two or more persons or corporations carrying on a business in common. The capital for a partnership is provided by the partners who are liable for the total debts of the firms and who share the profits and losses of the business concern according to the terms of the partnership agreement.
Your name must be unique, and not deceptively similar, to any other trademarked name or business. It is also required that your name not be used to intentionally misrepresent the products or services you offer. For LLCs, nearly all states will also require you to add a signifier of your limited liability status, such as "LLC" or "L.L.C." to the end of your company's name. You may be able to operate under a name other than your formal LLC name by applying for and using a dba.
The LLC is typically the best choice for smaller entities. The LLC structure provides a great deal of ownership flexibility in that an LLC may have any number of Members (owners) including non-US citizens and subsidiary companies. LLCs are also able to distribute several different classes of stock or ownership interest. However, their owners are typically required to pay a self-employment tax.
Where the equipment you use in your business was acquired as a gift, you may report your estimate of its current market value on the Business Property Statement (that is, what you think it would sell for in the open market place). Enter that estimated value in the most current year's cost line and add a note indicating that the entry is an estimate.