The kerfed lining is a strip of wood, with many saw cuts (kerfs) cut most of the way into it leaving only a thin strip of continuous wood. This makes it flexible, so it can be bent around and glued to the inside edges of the sides providing a gluing surface for the top and back.
The lining can be done in one of two ways. The ‘standard’ way is to have the kerfs facing out with the thin strip of wood glued to the side. The other is ‘reverse’ where the kerfs face inwards, and the thin strip of wood is facing out. I do reverse kerfing for two reasons. One – I think it makes a nicer, cleaner look inside the instrument to have that continuous strip of wood. Two – (this is the most important) – when reverse kerf lining is glued onto the side it makes the side very stiff. Before the lining is added, when the sides are just glued to the end blocks, the sides are springy and bouncy. After the reverse kerf lining is glued on the sides are totally stiff, with no bounce or flex at all. An amazing difference from the addition of some little pieces of wood.
I also like to do kerfed lining that goes with the sides. Darker for darker sides and lighter for lighter sides. I cut all my own kerfing to I have some selection of alternatives.