The router doesn't really intercept packets. A router will get a frame, strip off the frame to inspect the packet, then route the packet based on the destination address of the packet, based on what is in its routing table (dropping any packet for which is has no destination). The router will build a new frame for the packet for the network type to which the packet is being routed.
Routers route packets between networks, so a router will have at least two networks attached to it, but it may have many networks, and it may route packets between all the attached networks.
Based on your comment, you are interested in how a packet gets to a router in the first place.
Routers get packets from the frames sent to them. That could be from a broadcast or multicast, but a router will not normally propagate those. If it is a unicast, then it was sent directly to the router form another source. That is usually the source host or another router.
A host wanting to send a packet to a different host will mask its (the source) address and the destination address with its network mask. If the results are equal, then the destination is on the same network, and the frame for the packet is built with the destination host as the frame destination. If they are not equal, then the host knows that the destination is on a different network, and it builds the frame with its configured gateway (router) as the frame destination.
The router will strip off the frame to get the packet and read the packet destination in order to route the packet to the next leg of its journey.
It seems your question may be confusing routing and NAT. Those are really two completely different things. NAT (Network Address Translation) is changing the IPv4 address (source, destination, or both) on a packet. A router or firewall may be a convenient place to NAT (if you need NAT), but it is not a requirement of routing. NAT is used if you need to use private addressing on your network for traffic to communicate on the public Internet, or you need to communicate with a network that has addressing that overlaps your network addressing. Because NAT breaks many protocols, you should only use it if you must use it to overcome a problem.