Braving the Volcano is the second quest that Giovanni gives us when we first rescue him (the other one that comes at the same time is the one-part Giovanni's Workshop Boosts). Supposedly these are to help him brave the volcano, but when you finally reach the first volcano (Mount Cinder) you'll need a little bit more before you're ready to go in.
One other note: once you get Giovanni's House, you will want to start collecting Brass Ingots from it, and not stop for a long time. You will be needing tons of these things to complete all the Brass-related quests that Giovanni and Magnus will be giving you!
Once you complete this quest, Giovanni gives you another, called Stilling the Rumble.