a.k.a. @ reply, @username, @+username+message, @and the person name, twitter mention, twitter message
An @reply is a Twitter update that begins with @username. It is a way to say something directly back to another person.
People say a lot of things on Twitter and sometimes you want to say something back. It used to be that Twitter folks wanted to talk directly to each other so badly that they started using the @+username+message to designate their message as a reply to another person. Since people loved using it so much, Twitter made it an official feature.
On Twitter, you can start a tweet with @username and it will appear in the person's replies tab. You can also reply directly from someone's tweet.
This is different from a "mention", which is a Twitter update that contains @username in the body of the tweet. The folks at Twitter noticed people frequently search for their user name (@username) to find the tweets that mention their own user name anywhere in the tweet (rather than just the beginning). So now mentions are also included under the replies tab. If you include more than one person in your update and you use the @username format, that person will also see the update in their replies tab. All tweets containing @username are collected for you, accessible by a
link in your sidebar. Click on your user name to see all replies made
to you, and mentions of your user name.