Normally motherboard has it's own network card already, The build options usually have choice of adding extra cards for specific build purposes like running server on one card only or building a firewall or.. something else like someone not liking the brand that is on motherboard.
And yes, network card (motherboard one will 99.99% of time work just fine) would be connected to the router/modem to access internet, unless you choose to use wireless card instead. (99.9% of motherboards do not have wireless network cards)