Reservations aren't essential, but they're very easy to arrange and they cost nothing if you have a JR pass. I'd certainly recommend making reservations for any long journey that you take.

If you're using a JR pass to ride a train without a reservation and you find that the unreserved car is full, conductors should let you take an available seat in an unreserved car. However, it's best to wait until the train has left the station to attempt this. If you take an unoccupied seat in a reserved car when the train is at a station, there's a good chance that someone will show up with a reservation for that seat. Alternatively, the conductor may warn you that you're going to have to vacate the seat at the next station because someone reserved it from there.

For Shinkansen trains, information about which cars are reserved / unreserved / green is shown on the electronic displays above the platform. There are also signs at the entrances to every car that indicate reserved / non-reserved.

There are also labels on platforms (and hanging on wires above platforms) that indicate the type of car at each point on the platform. However, these can be very tricky to interpret, because many different train configurations use the same platform and you need to know which labels are associated with the train that you're about to catch.