Think of it this way--the router is the telephone operator.
No. Post office is much closer.
You want to send a parcel overseas. You have destination address, but no idea where it is physically, nor how to get there. You take the parcel to local post office. They don't know the final destination any better than you do, but they know the next post office "along the route" to deliver the parcel to. Parcel is carried from post office to next. Eventually, the parcel reaches a postman, who knows the recipient.
Traffic between devices connected to same switch does not require routing.
You have a letter to your neighbor. You write her name on it and leave a copy of the letter at every door. She will pick up the letter and the other neighbors will discard copies. Once you (switch) learn where she lives, you stop leaving unnecessary copies at other doors.
There is some routing though (even when network is down). A program in your computer creates a packet to be send. Your computer makes a "routing decision" and has usually three options:
1. Loopback. Destination is in the same computer.
2. Link-local. Shout it out via the network card. Destination is in the same subnet.
3. "Outside". Send to router. Note that the router is a link-local destination.