Email Registration | drupal.org

Allow users to register using only an email address, ie. without username. The user, once registered then may log-in using their email address and password for authentication.

A username is generated and assigned based on the user name part of the email address. Subsequent users who have the same name (email prefix) are numbered sequentially. For example, "johndoe@exampleA.com" would be assigned username "johndoe", "johndoe@exampleB.com" would be assigned "johndoe001" and so forth.

UPDATE 5.x An admin checkbox option has been added to make username the same as email. Note that this presents a security problem -- which is hard to solve. Please use with caution.

Username is the displayed name for posts, comments, etc and may be made changeable if so desired.
(Administer -> User Management -> Access Control) ... [X] change own username
This privilege allows a user to change their username in "My Account".

Profile module fields which are displayed on the register page no longer cause the Email Registration module to break.

Note:
You will probably want to change the welcome e-mail (Administer -> User Management -> User Settings) and replace instances of the token !username with !mailto