Both Verizon & AT&T (as well as some other providers) offer 2 different systems for your home.
Their
Home Connect system is a cell based method of tying your existing home phone network or a wireless (non-cell) phone to a cell based system. It does not require an existing internet connection, but does require the ability to connect to a cell tower. The basic unit does not provide data connections.
They also have a system that extends cell coverage using a device that is essentially a small "cell tower" tied to an existing high speed internet connection - the VOIP system MrWizard referred to. Verizon calls it
the Verizon Network Extender. This device allows you to use your cell phones in a location that does not have adequate cell coverage as long as you already have a high speed internet connection.
From the description of the original plan, it looks like a repackaging of Verizon's services. They have had a plan that adds their Home Connect service to their "Share Everything" plan for $10.00 per month (and 2 year contract). This looks like the same thing going in the other direction.