Bearer Bonds are bonds which are not registered to any individual, but are owned by the person who physically holds the bonds. Therefore, if you are the owner of a bearer bond, you must be extremely careful about making sure that the bond is not lost, stolen, or damaged. The opposite of a bearer bond is a registered bond.
These bonds are no longer issued in the United States, but are still issued in other countries. Some countries even still issue bearer stocks.
Bearer bonds are not technically illegal in the US based on my understanding, it is just that the court decision, Tax Equity and Fiscal Responsibility Act of 1982, and upheld in the case, South Carolina v. Baker, basically destroyed the reason for a corporation or governmental entity to issue bearer bonds.
The law basically said that any interest paid on bearer bonds issued by a corporation would not be deductible to the corporation, and the interest paid on bearer municipal bonds would no longer be tax exempt.
Bearer bonds are bonds which are not registered to any individual, but are owned by the person who physically holds the bonds. No name appears on the bonds and no record is kept of ownership of the bonds, unlike the registered bonds. Bearer bonds are issued with sheets of coupons, with each coupon representing an interest payment. The bondholder would take the coupon to a bank and cash it or deposit it like cash.
The reasons the US Government did this were twofold, but both relating to taxation in my opinion. First, for the corporate bearer bonds, individuals were not reporting the interest they were receiving, and the government had no way of tracking that information.
In regards to municipal bonds, normally held by the wealthy due to the tax exempt interest, families were avoiding estate taxes by keeping the bonds in a shared location with heirs. When the bondholder passed away, the heir would take possession of the muni bonds and the government would have no way of finding out about the transfer to the heirs for estate tax purposes.
More info available at Get Tax Free Interest from Municipal Bonds.