Verizon pre-announced its VoLTE service a few weeks ago, but now it's rolling out. Customers can finally start taking advantage of VoLTE calling starting today—that assumes people still make phone calls, which is debatable. You need to have a supported device of course, and there are still a few caveats.
VoLTE allows you to place calls entirely over 4G LTE, finally leaving that crappy old 3G connection behind. Call quality will be vastly improved, and you can make 6-way calls. Verizon's Advanced Calling 1.0 deal also includes native video calling over the LTE connection.
You can only use VoLTE if you've got a supported device, which right now only includes the LG G2 and Samsung Galaxy S5. Verizon says the selection will expand soon. Several of Verizon's promo images use the G3, so I have to assume an OTA update is in the works for that phone. While VoLTE is a standard across all carriers, VoLTE calls aren't cross-carrier yet. So you'll only get the higher voice quality when calling another Verizon customer with a supported phone.