Enum is basically a method of linking traditional pstn phone numbers to the DNS. For example the phone number of my old family home was 64 (4) 383 8249. This would be 22.214.171.124.126.96.36.199.4.6.e164.arpa as a Enum.
The Dom Post has an article on some Enum initiatives in NZ. As one can see there are many possible benefits with Enum, along with issues such as privacy to be worked through.