This is a telephone switching center, or central office, that does not connect directly to the customer. It instead provides the connections between the far-flung C.O.s, dealing with trunk lines rather than customer lines. It can connect offices in the same network, or can provide connections between networks. They are often located in the same facility as an end office.
It's at the tandem office that any call whose route spans two or more C.O.s is routed. Based on a table of address ranges, your call is sent to the appropriate end office, or another tandem office if necessary. In the 911 system, the tandem office is the facility where your call is tied in with your location.