Your home router tends to 'act' as both DNS server, and DHCP server.
DHCP server, when receiving request of IP address from connected devices, will provide a few things:
- IP Address
- Default Gateway (which will point to itself - aka the router - 192.168.1.1 in this sample)
- DNS Servers (which most home-use routers with default configuration will default to the gateway as the DNS server)
In a sense, this makes it easier for home user, they don't have to worry about the nitty gritty details of getting Gateway, and DNS servers, as all are provided by the DHCP server (aka router). And the router, acting as a DNS, simply relaying request through your ISP DNS server.
Your router will (most often than not) receive DHCP information from your ISP. DHCP information from your ISP will have similar info, but used by the router, not passed down to your PC. Hence onward, your router will have your "Public IP" address, Default Gateway (somewhere on the ISP), and DNS server information.
If you want to specify your own DNS server, you can do that in 2 places:
- Individual PC / Devices connected to the router
If you set it on the router (and everyone are still on "Obtain IP address automatically"):
Every devices will get the DNS server you set on the router.
If you set it on the PC / Devices connected to the router, the ONLY THAT DEVICE have that DNS server information on them, and all others will use whatever your router gave to them.
Hope this helps. If it is not clear enough please let me know and I'll try to add some more details, or some graph to make it easier to understand.