What are @Replies?
An @reply is a public message sent from one person to another, distinguished from normal updates by the @username prefix. If a message begins with @username, it is treated as a reply. You don't need to be following the person to reply to them - all of your replies are visible in the @username tab in your home page sidebar.
What are Direct Messages?
Direct messages are private messages sent from one Twitter person to another. You can only send a direct message to a person who follows you. When you receive a direct message, it is saved in your direct message inbox, accessible from the Direct Message tab in the sidebar in your home page.
In summary then:
An @reply is a public message sent regardless of follow-ship that anyone can view. A direct message can only be sent to someone you follow, and is a private message that only the author and recipient can view.
Ken Thompson (aka The BumbleBee) blogs about bioteams, virtual collaboration and business simulation at www.bioteams.com.