Bearer Share
  
Securities can be issued in two ways: registered, or bearer. These days, most are issued as registered, meaning that the business that sells the security keeps track of it.
With the bearer form though, there is no record of who owns the security, and whoever has the certificate in hand is assumed to be the owner and entitled to the payments on said security. The bearer share is an equity security issued in this way: the business who issued it holds no record of it. The company pays dividends to whoever presents the certificate. As you may imagine, this leaves a lot of room for conflict over who is claiming the share, which may be partially why this practice is dwindling.
The incentive here is that transfer tax may not apply (because the transfer is not recorded). The downside is that the share can be lost if you lose the certificate, which stands as proof.
Picture inheriting from a great-grandparent. The bearer share might simply be a certificate in a lockbox to be sorted out...whenever someone gets to it. You might benefit with the lack of extra tax, but then again, if you have a shifty cousin that gets to the box first, they could take your certificates and you would have a hard time proving it.