Until 1984, every phone call you made was anonymous. The person on the other end had to pick up to see who was calling. Now, every time you make a call, your phone company sends out a Calling Party Number (CPN) with the call. If you block the CPN, the recipient will not see your number on her caller-ID display. Instead, the call will be labeled as anonymous or from an unknown caller.
To have caller ID blocked permanently, call your local phone company and have it turned off. Some companies, like Verizon, allow you to deactivate the feature by going to your "account preferences" page on the web.
Know that some cell phones allow permanent caller-ID blocking through the main menu. Consult your owner's manual for instructions.
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To block caller ID on an individual-call basis, dial *67 (or 1167 on rotary phones). You'll hear an interrupted tone, whereupon you dial the number you want to reach.
If your number is permanently blocked but you want to get through to someone whose phone does not accept calls from such numbers, dial *82, then the number.