There are a couple of ways of doing this and it depends on how secure you need to be.
Use *67 to hide your phone number
If you're calling to bug an ex, beg for forgiveness or whatever, and you just want to hear them answer the phone before hanging up, you can just use *67 as a prefix on their number. This works with most carriers in the U.S., I believe. It just suppresses caller ID.
Use a pay phone
Then there is the call that you want to make to turn in your neighbor who is selling drugs. Just go somewhere and use a pay phone. Don't use a credit card. You could be found, but it's less likely.
Use a "free call" service
The Internet has also opened up another avenue. There are “free call” services that will let you dial a number for U.S. and Canada calls from a web page without any registration. Make sure the mobile device you are using has cellular data switched off to ensure it is on Wi-Fi and only Wi-Fi.
And remember, if you're doing illegal things, you're likely to get caught eventually.