You should choose a name for the partnership. For more information, read How to choose a name for your business. As a partnership, you do not have to register a business name but you need to publicise a fixed point of contact so that people connect the business name with you in your capacity as a business partnership (eg a statement on your business stationery: 'John Smith and partners, 234 High Street ...etc').
You must choose a 'nominated partner' who is responsible for administration and tax returns.
The nominated partner must register the partnership for self-assessment with HMRC via the Government Gateway. The other partners must register for self-assessment with HMRC. If you register later than 5 October in the second year of your business, you could be charged with a penalty.
To be clear about your relationship with your partners, you can consider drawing up a Partnership agreement. Read Setting up a partnership for more information.
For more information on the different types of partnership, read Types of partnership. For more information on your responsibilities when running a business partnership, read Running your business.