What is this called/Who do I need

Agree with others. Block up the hole for now. If the broken up tiled area projects in to the room, cover it with a bit of ply and a rug.

When the time comes, and I'm guessing you want to fit a free standing stove with an inglenook arrangement...



If/when you plaster the area around the stove, do not use plasterboard - It is not rated as "fireproof", the paper will char, and the plaster inside will crack and fall to bits. Do not use stuff like Thistle Multifinish or any other gypsum plaster as the heat will cause cracking and eventually, the stuff will fall off the wall.

Whilst you can get heat resistant plaster, it is a high alumina cement product and darned expensive. Lime plaster, done properly, would be ideal in the area. Or if you are happy with a rustic and slightly rough surface, cement board would hide the bricks.