You want to jailbreak your Verizon iPhone?  If you have just purchased a Verizon iPhone chances are you are running either 4.2.6 or 4.27.  In that case you are in luck.  Read carefully below as there are several different options depending on what firmware you have as well if you are on a mac or pc.

If you are running 4.2.6 then read below:
Download Greenpois0n RC6.1 Windows.
Download Greenpoison RC6.1 Mac.

If you are running 4.2.7 then read below:
Download Sn0wbreeze 2.6.1 Windows.
Download redsn0w Mac. **Only for 4.2.7 on mac, now head to Cydia and install the Verizon Untether for 4.2.7 from the repo.

There you have it, regardless of what your situation is you should be good to go with the 4.2.6/4.2.7 jailbreak on your Verizon iPhone.