A sole trader is a business started and ran by one person. You can work your own hours but you have full responsibility. You get to keep all profit.
A partnership business is a business ran by a group of people and numbers go between 2-20 people. Not as much money.
Private Limited Company:
A business between a family and friends, not publicly available for buying shares.
Public Limited Company:
Shares bought by the public to help business stay running.