"Fare" is the base fare for covering the distance. If you took a slower train or a local train, this is what you would pay.

"Charge" is the extra fee for taking a Limited Express or Shinkansen.

If you are travelling on a Limited Express or Shinkansen you pay the sum of the two, as you noted.

There are different rules that apply to these two -- for example, if you are travelling more than a certain minimum distance, you can make stopovers within a time limit (depending on distance) without paying more "fare." However, you have to pay the "charge" on a per sector as travelled basis, and the "charge" is definitely non-trivial even for short distances -- this is what keeps people from taking the Shinkansen from, say, Tokyo station to Yokohama.

Hyperdia is a very useful site. About the only thing that can go wrong is you need to spell the station names exactly in the form the site expects which isn't always obvious. In that case, rather than spending hours guessing, I pick a location BEYOND my destination and get any train. Then I click on the "timetable" icon and see how the intermediate station names are spelled by Hyperdia.