If over the years you've made a few tweets you regret and don't really care if the others are saved for posterity, you can delete all the tweets in your Twitter account with the TwitWipe web app. Of course you can just delete your Twitter account, but that will get rid of everyone you're following, your own followers and unlink your account from any apps you might still want to use. TwitWipe just gets rid of the tweets and leaves everything else intact.
If you want to wipe your tweets, just sign into Twitter, visit the TwitWipe link below and click on the "TwitWipe this Account" button. There will be a confirmation and a disclaimer click. Depending on the number of tweets you have to erase and the number of people simultaneously TwitWiping it may take a few hours for all your tweets to be annihilated.
Keep in mind there is no going back once you've wiped out your tweets. If you think you may want to save them beforehand go visit Backupify.