DBI is a database-independent interface for the Perl programming language.
DBD::mysql is the driver for connecting to MySQL database servers with DBI.
DBI is the basic abstraction layer for working with databases in Perl.
DBD::mysql is the driver for using MySQL with DBI.
Net::MySQL is a pure-Perl implementation of the MySQL client-server protocol. (It is not necessary when using DBI with DBD::mysql, but may be useful in environments where you are not able to compile the MySQL client library required by DBD::mysql.)