Hi Gitte,
From reading your question I think that you may be little confused by all the fares displayed in Hyperdia.
Takes this example: Kyoto - Tokyo. Which has the following ticket price: ¥ 13,020 consisting of the fare:¥ 7,980 and Seat Fee:¥ 5,040. The fare is the price you pay for the amount of distance traveled (km). The seat fee basically is the charge you pay for the Shinkansen and other limited express trains. Both fares make up the total price of the ticket. Lastly there's 3rd fee for seat reservations, this is what you pay for the actual reservation itself. In the case of Kyoto - Tokyo this is 310yen.
Normally you don't have to worry about these fees as the JR Pass covers them all, however it can be good to know about them if you want to find out if a JR Pass would be useful for you.
Lastly, the only trains which don't have a seat fee are local trains, which are mainly commuter trains.
Also see our blog post for more information about seat reservations and how to make them.
Hope this helps!
Daniel