A tarot box is a lidded box sized for a deck, usually wooden, often carved or with a drawer. The reason people buy one is almost always the same: the box the deck came in has split along the fold, which is what publisher boxes do after a few hundred openings.
Measure first. Decks are not standard.
There is no enforced size for a tarot deck. The common Rider-Waite-Smith format is roughly 70 × 120mm, but Thoth decks run larger, many indie decks are larger again, and "pocket" or "mini" editions are much smaller. Oracle decks are the worst offenders — some are nearly square, some are considerably oversized.
So measure the deck, not the reputation. Length, width and — the one people forget — depth. A 78-card deck on thick stock with gilt edges is noticeably thicker than a thin-stock one, and a box that fits the footprint but not the height is no use.
Sections, and what they are for
A single-compartment box holds one deck and whatever you put on top of it. A two-section box usually separates majors from minors, or deck from notebook. A four-section box is for someone who reads with several decks, or who keeps stones, a candle and a cloth with the cards.
Sections cost interior space. A four-section box of a given external size holds less than a plain one, because every divider is a few millimetres of wall. If you own one deck, plain is better.
Drawer boxes
A drawer opens without lifting anything off the top, which matters if the box lives under something on a shelf. It also means the cards slide against the drawer base every time, so a lined or smooth-finished drawer is worth having.
Or a bag
Worth saying, since we would rather you bought the right thing: a drawstring bag protects a deck perfectly well, costs less, travels better and does not care about deck dimensions. A box is for keeping a deck somewhere, a bag is for carrying it. Plenty of readers have both, and if you are choosing one, choose by whether the deck stays home.
Wood and finish
Most carved boxes are mango, sheesham or mahogany-stained hardwood. The finish matters more than the species — an unsealed interior will scent the cards over months, which some people like and others very much do not. If you want the cards to smell of nothing, look for a sealed or lined interior.