My bathroom has white caulking in some areas that is then covered by some brown stuff. Some of the brown stuff has broken off. What is this brown stuff called? It has a rough texture.

Grout?

Maybe? I thought grout was only used on tiles, and not something put on top of caulk. Is it something like this?

https://www.homedepot.com/p/Custom-Building-Products-SimpleGrout-380-Haystack-1-Qt-Pre-Mixed-Grout-PMG380QT/100676316

If so, how do you apply it like it's shown in my pic?

Hard to tell from the pic but grout will always eventually fail at change of plane or change of materials. These areas are best sealed with caulk. You can purchase sanded caulk that matches the grout pretty close. And grout should never be applied over caulk. Hard to tell in that pic but that could be sanded grout caulk on top of the white caulk. If so that is still incorrect to caulk over other caulk

The pic may be a little confusing, but the white at the top is a vertical marble bathtub wall, and the caulk is sealing the gap where it meets the bathroom floor which is tile.

Yes I think what JP76 is correct. You'll need to remove both the white caulk and the brown grout. Then replace with sanded caulk that matches the brown grout as well as possible.

Ok, so I didn't know sanded caulk was a thing. What I currently have must be sanded caulk on top of regular caulk. Why not just put more sanded caulk on top of the regular caulk? The white caulk is perfectly fine.

What I see is a piece of tile, and then where this piece of tile meats the tub, their is a joint. On one side this joint is filled with the grout that was used when the tile was installed, and on other side the grout has failed so white caulk was used. I would recommended cutting both out and, at minimum, caulk along that joint with white silicone caulk where it meets the bath tub.

Or if you want to make it look better and not so contrasted then just used the custom sanded grout caulk that you can buy at Home Depot and caulk along the entire bath tub with that and fill the joint. Either way, best to cut out existing grout and caulk so that you are doing the same thing along the tub at the same time, this creates a consistent look.

Because caulk over caulk will fail like it is doing now. Also if you zoom in on the pic I see caulk failure on the white caulk at the top where it meets the marble. The white caulk underneath looks like 100% silicone which is mineral spirit based and nearly every grout caulk I have seen/used while siliconized is usually water based. The 2 do not play well together.