The firm can issue (additional) capital by issuing shares. Shareholders own the firm; they bear the most risk, and have the highest (potential) rewards. Investors usually pay cash to become a shareholder, but in principle, it is possible they pay for the shares in other ways. For example, by transferring ownership of a building, providing services, or converting a bond.
When shares have special rights, they are called preferred shares, as opposed to ‘common’ shares. Shares that are listed on exchanges are (with some exceptions) common shares. Usually, ‘shares’ and ‘common shares’ are used interchangeably. The special rights of preferred shares can (for example) be related to hiring/firing the firm’s management (to frustrate hostile takeovers), having a preferred claim in case of liquidation, or receiving a (predetermined) dividend before the common shareholders do. I include a section on dividend-preferred shares.
For a corporation, the articles of incorporation will include the possibility for management to issue shares. The maximum number of shares a firm can issue is called authorized shares. The shares that have actually been sold to investors are called issued shares. Not all issued shares are held by investors, it could be the firm has purchased some of their shares back from the investing public. These shares are called treasury shares. The difference (issued shares minus treasury shares) is outstanding shares.