The number of shares of stock authorized refers to the total number of shares the corporation's Board of Directors has the authority to issue. This number is stated in the articles of incorporation (also known as the certificate of incorporation), and will generally be separately stated for common stock and preferred stock (if any preferred stock is authorized).
The corporation must take care not to issue more shares than are authorized under the articles of incorporation, or else an overissue will occur. Thus, if a corporation desires to issue more shares than are authorized, an amendment to the articles of incorporation is necessary. One might ask why corporations don't simply authorize a very large number of shares to avoid this possibility. One reason is that the shareholders may wish to constrain the board's ability to issue shares beyond a certain amount without their consent. Another reason is that franchise taxes are often computed partly or entirely on the basis of authorized shares, making that an expensive approach.
The difference between the number of authorized shares and the number of issued and outstanding shares is that issued and oustanding shares are those previously issued out of the total authorized and not repurchased or redeemed. In other words, the number of issued and outstanding shares is always less than the number of authorized shares. If the corporation would like to issue additional shares that would take its total number issued and oustanding above the number of authorized shares, the Board must amend the articles of incorporation, which requires a shareholder vote.