Which type of business is defined in part by its unlimited liability?

sole proprietorship

This is a business, unincorporated,run by an individual with no distinction between the owner and the business.  The owner is not only entitled to all the profits but also,in case  business goes wrong, is responsible solely for the losses. In addition, there is no distinction between the owner and the debt, such that in case the business goes into debt the responsibility for paying these debts rests on owner.