Often, trains have some wagons dedicated to reserved seats ('shitei-seki'), whilst others are dedicated to unreserved seats ('jiyu-seki'). This is the case for Shinkansen trains.
However, most local trains don't have reserved seats, whereas other trains like the Narita Express only have reserved seats. (This does not mean that you will need to book in advance but only that, when you buy your ticket, you will be assigned a seat number).
WHY SHOULD I BOOK?
If you are not catching your Shinkansen during peak hours (between 7 am and 9 am and 6 pm-8 pm), and during bank holidays, you do not need to book your seat and you can use any non-reserved seat by showing your Japan Rail Pass.
In fact, the Shinkansen trains run very frequently (every 30 min), particularly on the Tokaido line, between Tokyo and Kyoto, so you don’t have to worry about booking.
Starting from May 2020, a seat reservation near the luggage area will be required in the Tokaido, Sanyo and Kyushu lines if the sum of the dimensions of your luggage exceeds 160 cm.
HOW CAN I BOOK A SEAT?
However, if you want to book a seat you can do so without additional charges before you catch your train by going into a booking office at the train station (midori no madoguchi) and showing your Japan Rail Pass.
Booking website are usually in Japanese only, and therefore not very easy to use for non Japanese speakers. The JR East Shinkansen Reservation office is the only one with an English version that allows online booking.
However, you can only book Shinkansen and Limited Express trains running on the JR East network, that is to say Tokyo, the Northern part of Japan and the Narita Express.
Since the start of June 2020, it has been possible for Japan Rail Pass holders to reserve their seats for free at automatic ticket machines as well as JR ticket offices.