A Limited Company is registered with Companies House. It will have it's own Company Number too.
There will be a minimum of one Company Director and shareholder, although these can go up in number.
Sole Trader businesses are typically owned and controlled by one individual (although you still may employ staff).
They are not registered with Companies House.
Partnerships are typically owned and controlled by two or more individuals. They may, or may not, have additional staff employed too.
They can sometimes be a Limited Liability Partnership (LLP) which are registered with Companies House.