It's actually harder to explain than you may think.
The term dates back to the coin-op age. Basically, it's an increasingly difficult fixed length game mode, accompanied by a small story, either about the character you selected or just generally about the situation you were thrown into, or accompanied by no story at all. No extra content whatsoever, as in you would've just gotten in front of the coin-op cabinet, tossed in a coin and started playing.
Mostly used in fighting games or endless survivals (1 life, restart on lose, get as far as you can), It's a pick-up-and-play no-brainer mode where you can just fire up the game, press start and play; there's rarely any customization besides apparences, there are no quirks, no perks, no level ups, nothing else than the game's core itself.
The best example that comes to mind is, funnily enough, Pokémon: remove the story, the breeding, the role playing and the weakness/resistance system, even out the differences amongst them, give them a fixed moveset and you've got yourself something vaguely resembling an arcade mode.