Sole traders:
Sole traders are self-employed and all profits and losses made by the business are in total control of the individual's finances. The individual has to keep track of all finances involved in the business and file them in a yearly self assessed tax return, primarily focussing on income tax and national insurance (classes 2 and 4). A business name is purely optional and doesn't need to be registered if you're trading under your own name, however in the case of trading under a business name trademark regulations amongst others still apply.
Limited Companies:
Private companies limited by shares (LTD) are companies ran privately by a board of majority shareholders or an affiliated NGO instead of members of the public, and any stake in the company is bought and sold privately and the liabilities of shareholders in relation to the company only go as far as their stake in the company. Opposed to this is a public limited company (PLC) , which unlike LTD's can have shares bought by essentially any public entity. PLC's however must have at least £50k in value worth of shares. What defines a limited company in either context is the separation of the company's legal and financial assets from the individuals', and the profits after tax can be used to the discretion of the company. Another form of limited company is a company limited by guarantee, which is nearly identical to a company limited by shares apart from the business being ran by one or more 'guarantors' who put money into the business but don't gain profit from returns, and instead the money is put back into the business for future endeavours. This business model is usually reserved for non-profit organisations such as Charities, Community Interest Companies (CICs) and Community Benefit Societies (BenComs).
Partnerships:
A Partnership is similar to a sole trader business in that the individual is responsible for the businesses legal and financial assets, however in this case there are multiple individuals involved in the business. One of these partners can be an entity rather than an individual, such as a limited company. One of the partners has to be registered as the nominated party responsible for tax returns and business records.
Social Enterprises:
Social enterprises are businesses with a huge focus on reinvesting into the social sector to create positive change in various communities and the world at large. These goals could be tackling social problems - such as 'The Big Issue' which is trying to tackle homelessness as well as societies perception of the homeless - or even environmental issues, among a multitude of other issues. A Social Enterprise can be set up as: a Limited Company; a Charity or Charitable Incorporated Organisation (CIO); a Co-operative; a CIC or as a sole-trader or business partnership. Whichever one the enterprise sets up as is dependant on whichever option will help with their mission the most and how many assets they have available to them.
On top of the set-up requirements for a social enterprise, a CIC would require a 'community interest statement' stating the business plan for the enterprise, an 'asset lock' to put limits on profit made by an individual as opposed to re-investing into the business and a written constitution. Once all this has been submitted, the CIC Regulator has to approve the business.
Charities:
Charities are companies that benefit the community in some way shape or from with a clear charitable purpose. Not all companies that benefit the community are charities, and its this purpose that sets charities apart. To qualify as a registered charity individuals will have to go through the charity commission as opposed to companies house, and are required to provide proof of public benefit as well as a £5,000 annual income. Charities can never use income for private benefit, however some charities can still register as a limited company by guarantee, allowing them more options in terms of liability at the cost of having to register with both companies house and charity commission.
Unincorporated associations:
Unincorporated associations aren't required to be registered with the government or taxed for as long as they don't start creating profit for the individual. Due to its nature of being independently ran an association's financing will be solely down to the individuals who made the agreement to become associates. The reasons for making an unincorporated association could be to create a volunteer ran club for a local community, such as a sports team.
References -
The Arts Development Company - How to choose the right business model; What is a social enterprise?
UK Government: Sole trader; limited companies; Shareholders and Guarantors; partnerships; social enterprises; Unincorporated associations; Alternative charity options; set up a charity
ENSFC teaching blog - Types of company in the UK