Most private studio teachers are sole proprietors.
Your state may require a general partnership (hereafter partnership) to file and/or to pay fees. If it doesn't, I strongly advise you to see an attorney for a partnership agreement prior to starting your business. To save money, you can do it yourself with a book from Nolo Press or something similar, but you probably want to run it by a real attorney.
Beginning a corporation requires complex forms, filing fees, plus sizable attorney and accountant fees. There are also on-going paperwork and fees.
So why would any businessperson want to set up a corporation?
With a SP or a partnership, you are subject to unlimited -personal- liability. This means that if your business goes into debt, your creditors can seize your -personal- property. If someone sues you (say a student's Uncle Festus trips on a rug and injures himself) and your insurance coverage does not meet the amount of the liability claim, your personal assets may be taken and sold to complete payment. Pretty horrifying.
A person in an incorporated business technically has security from seizure of his personal assets. Only corporate assets may be attached to satisfy the claim. I said "technically" because this security - - called the "corporate veil" - - is being "pierced" more and more often, so don't opt for a corporate set-up because you think it offers 100% protection for your personal assets.
Now let me backtrack a bit to a limited liability company. It is a cross between a partnership and a corporation, offering the personal asset protection of a corporation but the flexibility of a partnership. These are fairly new legal entities, and if you are considering one of these as an alternative to a partnership, you definitely need to research the specific laws of your state. I will not discuss the LLC further here.
If you are a one-teacher operation, your options would be a SP or a corporation.
If you are in business with some other teachers, any of these legal forms would be a possibility, depending on whether the other teachers are your employees or not.
For a partnership, the same idea pertains. The money you and your partner(s) receive is called a "partner's draw" or a "guaranteed payment;" these are not deductible. A partnership's employees' salaries -are- deductible, however, just as with a SP's employees. It just happens that the employee can't be you!
With a corporation, the law considers the business a separate legal entity from you (which is why your personal assets are protected from a suit against the business). Therefore, the business can hire you as an employee and take a corporate tax deduction for your salary. You receive a W-2 at the end of the year and report your "pay" as you would any other salary (on the front side of Form 1040).
In a partnership, the situation is likewise, except that the partnership agreement may spell out a mechanism by which the deceased partner's portion may be bought out by the other partner(s).
The corporation is different. It has a life of its own. A corporation may be dissolved -only- if the shareholders vote to do it. If one of the shareholders (owners) dies, his shares pass to his heirs.
The sole proprietor or the partner can set up a Keogh plan or a SEP-IRA, which are tax-deferred plans. (In a tax-deferred plan, you own no tax on the money in the year you contribute it; you pay tax only when you withdraw it, presumably after your earning years - that is, retirement - when your personal tax rate drops because your earned income drops or disappears entirely. There is legislation afoot which would allow pre-tax contributions -and- tax-free withdrawals at retirement.)
These tax deferred plans are the equivalent to a corporate plan.