There are different types and sizes of firms in the UK economy. Types include public limited companies, privately-owned firms, start-ups, state-owned businesses, social enterprises, co-operatives, and partnerships. In terms of size, micro businesses have 0-9 employees, small to medium sized businesses (SMEs) have 10-250 employees, and large businesses employ over 250 people. The document also discusses business births and deaths in the UK economy.