The Japan Guide explains in which seats free seats are offered and in which:
Most Shinkansen trains offer both unreserved seats (自由 自由, jiyūseki) and reserved seats (指定 指定, shiteiseki) in separate cars. Only the Hayabusa, Hayate and Komachi trains at Tohoku Shinkansen and Hokkaido Shinkansen and the Kagayaki trains at Hokuriku Shinkansen are fully reserved and do not include unreserved seats.
If you are not sure, you can check the train you are taking for HyperDia.
If there is an "Unreserved seat" option under "Seat charge", this train has unreserved seats. Otherwise not. Please note that you also have to pay a "seat fee" for unreserved seats. This is because the "fare" is actually identical to a regular train and the "seat charge" is a combination of the "limited express charge" and the seat reservation charge.
If you are on vacation in Japan and traveling on the JR Pass, you can request a free seat reservation at the station ticket office.