Authorised capital is the maximum amount of capital which can be susbscribed by the shareholders. Suppose a company has authorized capital of Rs.5,00,0000/- and paid up capital of Rs.2,00,000 then it means maximum 50,000 number of shares of Rs.10/- each can be allotted to the shareholders.
Paid up Capital is the capital which are already subscribed by the shareholders. Continuing the above example, it means 20,000 share are already been subscribed/allotted to the shareholders and remaining 30,000 shares can be allotted in future if someone wants to become shareholder of the company. However if you want to allot more than 30,000, then you need to first increase your authorized capital and then paid up capital. Authorised capital can be increased to any amount at any time depending upon your requirement after paying ROC fees and Professional Fees as applicable.