Prisaz's got it.
Your STB's for TV will require a standard Coax feed from the ONT in order to bring the TV channels to them -- just like you're used to with cable TV from the other guys. If you already have coax in the house and it enters near where the ONT is located, then the technician will likely disconnect it from your old provider's network and connect it to the ONT.
In addition, your STB's require an internet connection so that they can get their guide data, etc. This is done using a technology called MoCA which essentially allows delivering ethernet over coax. The source of internet in your house is the router which Verizon installed for your internet service. So, the technician will also need to connect a coax (on the same coax infrastructure as the STB's) to your internet router. MoCA is a multi-node technology and not point to point -- so as long as their is a path via coax from every STB to the router, you'll need good to go. Worthy to note is since your router is presently not connected to coax (it's using ethernet you said), a new connection to the router will need to be made, so you'll want to make sure it's located near a cable outlet.
Lastly, the router needs to be able to get to the internet. Today, that is being done via ethernet cable. If you have the right model ONT, as Prisaz mentioned, the technician can reconfigure the router so that it uses the Coax (a different frequency than the MoCA LAN the STB's use) to get the internet as well -- thus removing the need to have an ethernet cable run to the router (some like it seperate, others like me like it on integrated such that if I ever desire to relocate the router, I need only make sure I have a coax connection available nearby). Technically, there is no different between the MoCA WAN and Ethernet connection paths to the ONT, so either way is fine. If you have a preference, you should ask the technician to honor it.
As prisaz mentioned, there are different versions of ONT. Some models don't support MoCA -- they do internet on ethernet, TV signals on Coax, and telephone on phone wire only. In those scenarios your router must have both coax and ethernet run to it. I the other models like I have, the ONT supports MoCA and the only thing which needs to be run to the router is coax. In still other scenarios, such as apartment buildings, an entirely different techology is used employing frequency filters and vDSL to distribute internet and TV to you. So, the specifics may vary based on exactly what you have installed.
