Define the term unlimited company?

A business can be carried on by an individual (as a sole proprietor), a partnership, or a corporation.
The word 'limited' means that the liability of a shareholder for the unpaid debts of a corporation is limited to the amount of money the person has invested in the corporation. The person can lose his or her investment if the corporation 'goes under' but the creditors of the corporation can't make the shareholder 'pony up' for any more.
The word 'company' simply indicates that there is more than one person involved. It is an imprecise word that is in general everyday use, and has no special meaning in corporate law. It does not, by itself, indicate a limited liability corporation. Frankly, it would be best to avoid the use of that word, since its meaning can be misleading.
When you referred to 'unlimited company' you probably intended to refer to the concept of 'unlimited liability.' If a business which is being carried on as a sole proprietorship or a partnership is not incorporated, and 'goes under,' the proprietor or partners are personally liable for the debts of the business.