A company is a legal body in its own right, unlike a sole trader. Directors and shareholders may change, but the company will continue until it is legally wound up. A private limited company (which a small business would be) must have at least one shareholder and one director, plus a company secretary.
If you are a sole trader then you are the sole owner of the business. Whilst the business can still employ other people, you do not share ownership or responsibility in any way. There is much less formal activity involved in establishing this type of business and it is probably the most common form of business start-up.
The Partnership Act 1890 defines a partnership as ‘the relationship which subsists between persons carrying on a business in common with a view of profit’. Partnerships may employ other people, but cannot normally have more than 20 partners. (Professional partnerships such as accountants and solicitors are exceptions to this rule.)
A social enterprise is a business with primarily social objectives, whose surpluses are reinvested in the business or in the community, rather than being driven by the need to maximise profit for shareholders or owners.
Kent based Shell LiveWIRE Grand Ideas Award winners in January 2009, Aquaread talk about the importance of partnerships to their business.