There are basically 3 ways that you can set up your own business. These are as:
A sole trader
In partnership with one or more people
A limited company
Each has its own advantages and disadvantages.
This is the simplest form of business model. There are no legal formalities required to set yourself up in business as your own boss. The downside is that you also carry the can if things go wrong. You are personally liable for all the debts of the business, and creditors could look to your house, car, savings etc. to settle any money owed to them. You do not have to produce audited accounts, but you will need to keep records for the taxman. If you do set up in business on your own in this way you should advise your tax office, as you will now be taxed as a self-employed person.
If you set up your business with one or more people, then you will be in partnership, whether or not you actually get anything down on paper - even if the partners are other members of your family. Your partnership will be governed by the Partnership Act 1896. Generally this is not what you want or need. Anybody working together in partnership (including families) should get a Partnership Deed drawn up, which covers how the partnership will work on a day-to-day basis, how it can develop and how it can be ended. Generally each of the partners is personally responsible for all the businesses liabilities on a joint and several basis, whether they knew about them or not. Therefore choose your partners carefully. Partners remain self-employed and only need to prepare accounts for tax.
This is the most formal way to run a business, and it has the added advantage of limiting your liability to the amount of share capital you have invested in the business. Note that Banks and others will therefore inevitably ask you for personal guarantees for loans etc. The disadvantages are that there is a cost involved in setting up a company - you can buy one off-the-shelf for less than £100. Also you have to comply with Companies Act legislation - accounts, records and returns at Companies House. As a Director you are no longer self-employed, but employed by the company, so you are likely to have to pay National Insurance and be taxed as an employed person.