The maximum number of shares a corporation is legally allowed to issue, as set in its Articles of Incorporation.
Authorized shares are the ceiling on how many shares a company can ever issue. This number is set in the Articles of Incorporation when the company is formed. You can increase authorized shares later, but it requires a board and shareholder vote and filing an amendment to the Articles.
Most Delaware C-Corps authorize 10 million shares of common stock and 10 million shares of preferred stock when they incorporate, though the exact numbers vary. The number of authorized shares itself doesn't mean much — what matters is how many are actually issued. A company can authorize 100 million shares and issue only 10 million; the other 90 million are unissued and don't affect ownership or dilution calculations.
If you're running out of authorized shares because you've issued more than you planned, you can increase the authorized number through a board and shareholder vote. This is a routine corporate action but requires paperwork. It's generally better to authorize generously at formation to avoid needing to do this early.