What will I need to repair the grout in a small section of my bathroom?
You can remove the existing grout with a grout saw. Your local hardware stores should sell one.

The catch here is re-grouting. You can buy pre-mix grout in pints at some big-box stores, but be prepared for it to not match your existing grout. You may need to saw all the grout out and redo it afresh.

I would remove the toilet first, however. Make sure that tile isn't loose because of the toilet, or improper mounting