Jaén is the world capital of olive oil — the province produces more of it than the whole of Italy. The local cuisine uses it openly: drizzled on toast for breakfast, slicked over salads, used to fry whole everything.
Two regional dishes worth seeking: pipirrana (a finely chopped salad of tomato, pepper, onion, and oil, sometimes with tuna), and andrajos (literally "rags" — a hearty stew of rabbit or hare with squares of broken dough). And of course gachas (fried-flour porridge) and migas in winter.
The city has a real tapas culture with the free-tapa-per-drink tradition still going. Stand at the bar, order a beer (cerveza), and a small plate appears next to it. Three drinks roughly equals dinner.