In addition to allowing you to send mass 140-character "tweets" to those who follow you, Twitter affords you the option to communicate with others directly through its "Direct Message" utility, where you're also limited to 140 characters. If you want to cease communicating with someone on Twitter -- and prevent her from seeing your tweets -- block that person.
How Twitter Works
On a fundamental level, Twitter allows you to communicate with others in three ways: sending tweets out to your entire social network, "tagging" specific users in tweets and sending direct messages to specific users. Although the first option allows you to transmit information only to other users who have willfully followed you -- and to anyone who happens to stumble upon your page -- the other two allow you to communicate with any user on Twitter, as well as for any user to communicate with you.
Just as MySpace and Facebook do, Twitter allows you to block users with whom you no longer wish to communicate. To block a user, visit his profile and click the "Gear" icon located just under his name. Selecting the "Block" option removes him from your followers -- if he follows you, it prevents him from mentioning you in his tweets and removes his ability to send you direct Twitter messages.
Several limitations to blocking on Twitter exist. First and foremost, the user you've blocked can simply log out of her account, visit your page and see all the tweets you've sent. If she wishes to communicate with you directly, she can simply register a new Twitter account -- something which takes less than a minute -- and follow you to see your tweets. At this point, it's also possible for her to send you direct messages again and to mention you in her tweets.
Protecting Your Account
One way to prevent blocked users from viewing your tweets and sending you messages using new screen names is to protect your account, which requires that you manually approve every user who wishes to communicate with you. Protecting your account is simple -- visit your "Account Settings" page and select the "Protect My Updates" option. From this point forward, only users whom you follow will be able to communicate with you on Twitter.