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The legal structure of your business is very important as it is a function of liabilities and tax deduction. The most common structures are:

Sole trader
Partnership
Company
Trust

Sole Trader
When you are a sole trader you are personally responsible and liable for the business and its activities. Commonly you may start out as a sole trader, and as you expand, you will need to seek out partners or investors. Your accountant is the best person to advise you if this is the best structure for your business.
Partnership
This arrangement is where you and your partner are liable for anything related to your business. It is a very good idea to get a solicitor to write up a partnership agreement to avoid disputes down the track.
Company
This is a legal entity liable for the debts it incurs and paying tax on income. Your accountant can help you set a company up, or you can buy one off the shelf, which may be more cost effective.
Trust
Depending on your personal circumstances, and the country you live in, trusts can be useful in business and estate planning. They can be quite complex, and again, your accountant is the best person to speak too.