We have a pretty large granite countertop in our kitchen. The main
part constists of 3 pieces of stone. There seems to be some type of
grout sandwiched between the pieces. The problem is that due to
settling of an addition to the house right under one of these
junctions, the grout has come out of that seam. Can one use any type
of grout or is there a special type of material used for this
application?

Thanks

You should use the same kind of material as the original installation.
It may not be grout, which is basically concrete. It may be a good
grade of silicone (not just latex) caulk.

Make sure the surfaces on both sides of the gap are exactly level. If
not, you will need to make them level by shimming.

The installation is about 11 yeas old so I have no idea what was
originally used. It is not a flexible caulk of any kind. The pieces
that I am pulling out have broken and a quite hard.