The private one
Your router hands each device on your network an address from a range reserved for exactly this purpose: 10.0.0.0/8, 172.16.0.0/12 or 192.168.0.0/16, set aside by RFC 1918. These addresses are not unique in the world — millions of homes use 192.168.1.1 for the router — and no network on the public internet will route them. That is the point: they only have to be unique inside your own house.
This is the address your operating system shows you. It is the right one for reaching your printer or your NAS, and the wrong one for anything involving the outside world.
The public one
Your router holds a single public address, assigned by your provider, and it translates between that and your private addresses using NAT. Every device in your house therefore appears to the internet as one address — the one shown at the top of this site.
This is the address a site logs, a game server sees, a mail server checks against a blocklist, and a firewall rule elsewhere allows or denies.
Carrier-grade NAT: a third case
Because IPv4 addresses ran out, many providers — mobile networks especially — now do the same trick a second time, putting thousands of subscribers behind one public address using the 100.64.0.0/10 range internally.
If your address falls in that range, your router's 'public' address is not really public either. You cannot host anything reachable from outside on it, port forwarding does nothing, and any blocklist entry against the shared address is about your provider's other subscribers as much as about you.
How to tell which you are looking at
If it starts with 10., 192.168., or 172.16 through 172.31, it is private. If it starts with 100.64 through 100.127, it is carrier-grade NAT. 127.0.0.1 is your own machine talking to itself. Anything beginning 169.254 means the device failed to get an address at all and made one up.
Everything else is public — and paste any of them into the lookup on this site and it will name the range and cite the RFC that defines it.