Fully paid up shares are those for which no outstanding amounts are due. All monies due to the company for the equity it has issued have been paid in full.
For example, a company which has issued shares to the value of £100 has received the full £100 for them.
Persons holding fully paid-up shares in a limited company can not be asked or required to contribute additional amounts should the company be liquidated and be unable to satisfy all of its debt.
Limited liability protects their personal wealth.