Twitter allows you to publicly broadcast updates that anyone can read. You also have options for sending private tweets that only people who follow both you and the recipient can read. These private tweets are called "direct messages." Direct messages are only visible to the accounts that send and receive them, and they do not show on your profile. Viewing private tweets is simple and can be done from the Twitter home page.
Direct your Web browser to Twitter.com and log in to your Twitter account.
Click on your "Home" button, next to the search field, then click on the "@Mentions" tab. This displays all of the tweets you've received that other Twitter users who follow both you and the message author can see, but the rest of your followers do not.
Press the "Messages" button at the top of the page, between the "Profile" and "Who to Follow" buttons. Click on the Twitter username of the person whose message you want to see. This will show all of the private Direct Messages you have received from that person.