Remove the old grout around the base of the toilet with a hammer and blunt chisel. Work carefully to avoid accidentally chipping either the toilet or the tiles. If the grout is already cracked and chipped, removing it is simple. If the grout is more stubborn, use a rotary tool with a grinding bit.
Dispose of the large chunks of grout, and vacuum up the small chunks and dust.
Mix half a cup of grout with water in a pail according to the instructions on the packaging. Use a stir stick, long screwdriver or other long instrument to mix the grout and water together.
Dip a putty knife into the grout after you've mixed it to the correct consistency, and apply a thin layer of grout to the gap at the base of the toilet. The job will be messy. After you have applied the grout, run your finger along the gap to smooth out the grout and remove any extra grout. Wipe up extra grout with a damp rag.