There are different business ownership structures, so it is very important to choose the right one for you and your business.
Here are the most common structures:

 Sole Proprietorship

Owner remains personally liable for lawsuits filed against the business
No state filing required to form a sole proprietorship
Easy to form and operate
Owner reports business profit and loss on their personal tax return
A sole proprietorship is a one-person business that is not registered with the state. You don't have to do anything special, you create one just by going into business for yourself.
Legally, a sole proprietorship is inseparable from its owner -- the business and the owner are one and the same.
Owners report their share of profit and loss in the company on their personal tax returns Schedule C

Sole proprietorships and partnerships make sense in a business where personal liability isn't a big worry

 Partnership

Partners remain personally liable for lawsuits filed against the business
Usually no state filing required to form a partnership
Easy to form and operate
Owners report their share of profit and loss in the company on their personal tax returns

 A partnership is simply a business owned by two or more people.  You don't have to file any paperwork to form a partnership -- the arrangement begins as soon as you start a business with another person.  The partnership's owners pay taxes on their share of the business income on their personal tax returns and they are each personally liable for the entire amount of any business debts and claims. 

Limited Partnership

A Limited Partnership provides some protection from liability for certain partners. Limited partners contribute capital to the business but cannot participate in its management. These limited partners have limited liability for business debts and obligations. General partners manage the company and have unlimited personal liability for judgments against the business.

Limited partnerships are costly and complicated to set up and run, and are not recommended for the average small business owner. Limited partnerships are usually created by one person or company (the "general partner"), who will solicit investments from others (the "limited partners").

The general partner controls the limited partnership's day-to-day operations and is personally liable for business debts (unless the general partner is a corporation or an LLC). Limited partners have minimal control over daily business decisions or operations and, in return, they are not personally liable for business debts or claims. Consult a limited partnership expert if you're interested in creating this type of business.

 Limited Liability Company (LLC)

Independent legal structures separate from their owners
Help separate your personal assets from your business debts
Taxed similarly to a sole proprietorship (if one owner) or a partnership (if multiple owners)
No limit to the number of owners 
Governed by operating agreements

The LLC (Limited Liability Company) is the newest form of business entity, and it is rapidly becoming one of the most popular ways for new and small businesses to operate. It is often described as a hybrid between a corporation and a partnership.

It limits personal liability, like a corporation, while providing a simpler and more flexible structure than most other forms of business. A single member LLC is taxed on the federal level similar to a sole proprietor or Sub-S Corporation, while a multimember LLC is taxed on the federal level similar to a partnership or Sub-S Corporation. Since a huge number of small businesses are sole proprietorships and partnerships, an LLC is frequently an excellent choice for these types of businesses. It retains most of the benefits of a sole proprietorship or partnership with the added benefit of limited liability.

 Corporation (for-profit)

C Corporations

Independent legal and tax structures separate from their owners
Help separate your personal assets from your business debts
No limit to the number of shareholders
Taxed on corporate profits and shareholder dividends
Must hold annual meetings and record meeting minutes

S Corporations

Independent legal and tax structures separate from their owners
Help separate your personal assets from your business debts
Owners report their share of profit and loss in the company on their personal tax returns
Limits on number of shareholders, who must be U.S. citizens or residents
Must hold annual meetings and record meeting minutes

Forming and operating an LLC or a Corporation is a bit more complicated and costly, but well worth the trouble for some small businesses. The main benefit of an LLC or a Corporation is that these structures limit the owners' personal liability for business debts and court judgments against the business.

An S Corporation is a special form of corporation (Note: The "S" in S Corporation refers to subchapter S of the tax code). S Corporations are based on C Corporations but they are not treated as a separate tax entity as C Corporations are. Instead, the income of an S Corporation is "passed through" to the personal income of its owners (shareholders) in proportion to their ownership interest.

An S Corporation is created by forming a traditional C Corporation and then filing the IRS Form 2553 (The Subchapter S Election) for federal recognition of S Corporation tax status. While the S Corporation has many of the same features as a C Corporation, there are some important differences.

Note: While the S Corporation features similar pass through taxation to an LLC, in the area of self-employment taxes an S Corporation can have an advantage over an LLC. The compensation (salary and bonuses) of S Corporation shareholders is subject to self-employment tax, but not on the profits automatically allocated to them as a shareholder. This can be an advanced and aggressive tax strategy, so be sure to consult with the appropriate tax and legal specialists before pursuing it.

 Nonprofit Corporation (not-for-profit)

A nonprofit corporation is a corporation formed to carry out a charitable, educational, religious, literary, or scientific purpose. A nonprofit can raise much-needed funds by soliciting public and private grant money and donations from individuals and companies. The federal and state governments do not generally tax nonprofit corporations on money they take in that is related to their nonprofit purpose, because of the benefits they contribute to society.

Cooperative

Some people dream of forming a business of true equals -- an organization owned and operated democratically by its members. These grassroots business organizers often refer to their businesses as a "group," "collective," or "co-op" -- but these are often informal rather than legal labels. For example, a consumer co-op could be formed to run a food store, a bookstore, or any other retail business. Or a workers' co-op could be created to manufacture and sell arts and crafts. Most states do have specific laws dealing with the set-up of cooperatives, and in some states you can file paperwork with the secretary of state's office to have your cooperative formally recognized by the state. Check with your secretary of state's office for more information.