Bearer shares are corporation stock certificates which are owned simply by the person who holds them, the "Bearer".
When corporations first came into existence, most shares were bearer shares. If you wanted to protect your interest in the corporation, you had to protect your bearer share certificates. To protect against theft and fraud, corporations starting keeping a register of the owners of the bearer shares which were issued, and notice had to be sent to the secretary of the corporation to record the change in ownership. Eventually, the corporation's stock ledger determined ownership, and shares only facilitated the transfer of ownership (and, indeed, today few people ever see the stock shares they own). Eventually, most U.S. states even dropped the provisions allowing bearer shares.