A bearer bond is the “ancient” type of bonds, rarely issued today. The idea was, someone would buy a bond in the market(let’s say for $1000), and it would be mailed to him physically. On the bond were “coupons” that had dates on them, which could be clipped off and deposited in a bank on the dates listed. These were the semi-annual interest payments. When the bond matured, you would take the actual bond (and any coupons you forgot to deposit) to the bank and get your $1000.
The problem with bearer bonds is the security risk. Anyone with that coupon gets the money. As far as I remember, there were no serial numbers “back in the good old days.” Today, most bonds are book-entry style, which means a transfer agent with a computer has a list of who owns the bonds, and the interest payments are automatically dropped into your brokerage account.