The DM is essential
It stands for "direct message" and is the equivalent of a Facebook message: Wall posts and tweets are public, but DMs are not. Anything you want only the intended recipient to see like, say, indiscreet pictures sent to a lover should be relegated to DMs instead of your timeline.
Do not Tweet what you can text
It is a public Website, and you will not put your text messages online, would you? Twitter is for networking, not for your nightly back and forth with that Amazon in accounting. If you delete a tweet, it won't necessarily disappear from the internet. This is doubly so for pictures, which are hosted by third party sites.
Go private, with reservations
The best way to prevent prying eyes from reading unscrupulous tweets is to go private. This protects your 140 character missives from anyone you do not want reading them. Be warned: Anyone with access to your timeline can take a screenshot. Nothing is private on the internet.
Turn off phone notifications and e-mails, too
If you use Twitter on your phone, by default the application will send you a text alert when ever you are tweeted. Turn this option off to avoid embarrassment. Similarly, you can never fully ensure the security of your e-mail account, and Twitter e-mails you every time you receive a message. Cover your bases and sleep with peace of mind.
Keep tabs on yourself
Search your name and Twitter handle to find out who's mentioning you on the service. It is not narcissistic, it is smart: Know what people are saying about you and you can put out fires before they get out of control.