Shinkansen has reserved seats, ordinary reserved seat and green reserved seat.
For some Shinkansen trains, there are unreserved ordinary seats. However, majority of seats are ordinary reserved seats. Therefore, if you take Shinkansen in popular lines, such as Tokaido Shinkansen, Sanyo Shinkansen, and Hokuriku Shinkansen, it is better to reserve your seat.
You can reserve a seat of Shinkansen which is covered by your JR Pass. (You can not reserve a Green reserved seat with your JR Pass ordinary seat.)
In order to reserve a seat, you need to go to a JR ticket office at JR stations. All stations which Shinkansen stops have ticket offices. You do not have to go to a JR ticket office of your departure station. You can reserved a seat at any JR ticket office.
At the JR ticket office, you tell your itinerary with your JR Pass. You will receive a reserved seat ticket showing your train number and departure and arrival time/station, your cart number, and your seat number.
For summer holidays (the middle of August), the end and beginning of the year, and weekends, Shinkansen may be very crowded. Therefore, if you travel on those periods/days, it is recommended to make a reservation as soon as possible (as soon as you arrive in Japan).
For non-busy time, you can reserve a seat just before your departure time. However, JR ticket offices are often very crowded, it may take more than 30 minutes to make a reservation. It is recommended to be at your departure station in advance with plenty of time.
For JR Pass, please check here.
For information about Shinkansen, please check All About Shinkansen.