If termites have eaten through part of a joist under your floor, the first issue is to stop the infestation, by hiring an exterminator. Then you must fix the damage, since those joists hold up your floor and can't have their strength compromised. The usual approach is to install a "sister" joist alongside the existing joist, replacing the strength that the old joist has lost. This approach works for a joist that's partly but not completely eaten through and hasn't started sagging (otherwise, consult a professional).
Things You'll Need
Pressure-treated joist (usually 2-by-10s)
3-inch galvanized nails
Access the joist from under the floor. Use your hammer and wood chisel to knock away any loose, soft wood where there was termite damage. Remove as much of the damaged wood as possible so that the remaining wood is solid.
Measure the length of the joist, from end to end, using a tape measure.
Cut the sister joist to the same length, using your circular saw. If there are any center beams or other obstructions along the length of the old joist, measure and mark them on the new joist and cut them out with your jigsaw.
Hold the new joist in place, with a helper, so it is sitting flush against the side of the old joist. Position it so the new joist is sitting on the edge of the support column or the shelf in the stone foundation, alongside the old joist.
Hammer 3-inch galvanized nails through the side of the new joist and into the old one. Put nails along the top, middle and bottom of the width of the joist, every 18 top 24 inches (except where the old joist is damaged).
Wear eye protection when cutting the new joist and hammering it in place.