I am confused about why the router allows both client and server and how to configure them. Do you need a VPN service like NordVPN?
It allows both client and server because you can set up a VPN client or a VPN server. Generally speaking, there is nothing that would stop you from setting up your own VPN server if you want – you do not have to be an ISP or a large company, you just need the right software, which your router has. (And with WG in particular, there is almost no difference between clients and servers.)
So you could say you would be making your own VPN service like NordVPN – a VPN tunnel protocol such as WireGuard or IKEv2 or OpenVPN or PPTP is the main building block for that.
But the job of VPN isn't really to "change your address"; it's to connect you to a distant network, like a very long Ethernet cable. When you connect to a commercial VPN service like NordVPN, it's not WireGuard itself but the actual VPN provider that "changes your address" by letting you remotely use their routers/gateways – the job of WireGuard is only to carry your packets to the VPN provider's network.
Not all VPNs exist for the purpose of Internet access; some might be used only for accessing remote services at home, office, or university. Generic VPN protocols and clients don't actually enforce that all traffic should go through the VPN – as with physical connections, your router's routing table decides what goes over the VPN connection.
In conclusion, you can use the router as a VPN server if you want to access your home PCs or other devices from elsewhere (there's a generic "WireGuard" app for every OS). And if the clients are then configured to route everything through the VPN tunnel, they'll be able to access the Internet using your home IP address as well.