This isn't an Android thing though, it's a network thing. Either your network supports anonymous calling or it doesn't. If it does, then some handsets can turn this feature on through one of the menu options, BUT, all that does is simply dial a network code or prefix number to tell the network to hide your number.
The GSM code is usually #31#XXXXXXX where XXXXXXX is the number you want to dial. Or to make it display *31#XXXXXXX. To check the status of it just dial *#31# on its own. You can often turn it on or off for all future numbers by dialling just the code on its own without the XXXXXXX.
If you're on a CDMA, tdMA, or iDEN phone it's usually *67XXXXXXX to hide, *30#XXXXXXX to display, and *#30# to check.
While the above work on most GSM and CDMA networks, the code is network specific. Try calling your landline or another mobile to see.