Kiln furniture is the shelving and support equipment used to stack ware inside a kiln during firing, rather than firing everything piled directly on the kiln's own floor.
Kiln shelves, flat slabs usually made of cordierite or another thermal-shock-resistant refractory material, rest on posts (short refractory columns of varying height) to create stacked, evenly spaced layers inside the chamber. That stacking is what lets a single firing hold far more work than one layer of ware sitting on the floor alone would.
Shelves are coated with kiln wash, a thin refractory slurry, often mostly alumina hydrate and kaolin, so that a glaze drip running off a piece during firing does not permanently fuse the piece to the shelf. Kiln wash is meant to flake or scrape off along with whatever it caught, rather than bond to it, and is reapplied periodically as it wears down or as drips accumulate on it. Bare clay contact points on a piece, such as an unglazed foot ring, are typically left unglazed for the same reason: even kiln wash cannot fully guarantee that a glaze run will not stick.
Shelves and posts are themselves a real investment and take up real kiln volume, so how a kiln is loaded, shelf spacing, post height, leaving enough clearance above each piece, directly affects how much a single firing can actually hold. A cracked or warped shelf is usually a sign it was thermally shocked (heated or cooled too quickly for its own refractory to handle) or loaded past whatever weight it was rated for, and a compromised shelf is a real risk to whatever ware sits on it if it fails mid-firing.
Not all shelves are the same refractory. Cordierite has the lowest thermal expansion of the common options, making it the most thermal-shock-resistant and the most forgiving of a fast temperature change, but it is comparatively weak and poorly suited to wood or salt/soda firing, since sodium and potassium vapor attack it and shorten its life. Silicon carbide shelves conduct heat far more efficiently (roughly ten times cordierite's rate) and tolerate higher top temperatures, but they oxidize over repeated high firings and cost more. Mullite and cordierite-mullite blends sit between the two: not as tough as silicon carbide, but immune to the vapor attack that rules cordierite out of wood/salt work, which is why they show up often in wood-fired studios specifically.