A partnership is the relationship of two or more 'partners' carrying out a business with a view to making a profit. You and your partners are responsible for running the business. You share profits between yourselves. You and your partners are personally responsible for paying the bills (apart from LLPs). Partnerships are not a separate legal entity (apart from LLPs).
A partner doesn't have to be an actual person, for example, a limited company counts as a legal person and can also be a partner in a partnership.