Sole Proprietorship: This type of business does not make a legal distinction between the business owner and the business itself. Although costs can be low (from taxes to licensing fees)the down-side is that you will be personally responsible for any business debt that may occurs, potential law suits, and your personal income tax reflects your entire business' profit.
A General Partnership is the same as a sole proprietorship but with several owners. Again, the set-up costs are low and there are relatively few regulatory requirements. And, again, the owners are completely liable for all the actions and debts that stem from the business.
Limited Liability Partnership(much like a LLC) is very similar to a general partnership. However, there is a separate classification of partners making each person's liability limited. The downside to this classification is it only "limits" a partner's liability, not erase it completely- which is a common myth. Also the filing and administrative requirements are complex.
C-Corporation is only a form of doing business that creates a separate legal entity from the individual owners. This legal entity can act and do business on its own just as a person would do (i.e. borrow money, enter into lawsuits and contracts, etc.).
With a C-Corporation your "owners" are now shareholders who are not personally liable for the actions and debts stemming from the company. However, if a shareholder wants to leave the business then the paperwork involved isn't easy, making a separation a long-one- which in the case of a nasty separation only makes the situation worse. Also, owners (shareholders) are taxed twice.
S-Corporation is a type of corporation but the business is taxed as if it is a sole proprietorship. Its formation is subject to certain legal criteria such as a maximum number of shareholders. The benefits, unlike a C-Corp., include owners only being taxed one time. However, the administrative costs are higher and all owners (shareholders) have to be U.S. citizens.
Many business owners either chose to set-up as an LLC or an S-Corp. Note, I will not advocate for either because you need to discuss your personal situation with a lawyer and a CPA. However, I will state that an LLC may have to pay more self-employment taxes than S-corps due to IRS regulations.
Have other business questions? Please ask as I am happy to answer them!