What does 'arcade mode' mean when gamers are talking about current games

Solution
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...

Vynavill

Honorable
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.
 
Solution
I gotta admit Vynavill explained it really well. If there is one thing Id add to that is that Arcade Mode has changed over the years, not to mention it means different things in different games, so Its not a defined term that specifically points to one thing or another.
Mostly it is used however nowdays when there is more than one mode in a game (If you look at Witcher 2 for example, there is a ARENA mode and new game. You could say ARENA mode is actually arcade, but it has not been called that way).

So its basic use as far as i see it is to easier comunicate with other players when you want to specify a "mode" you played on in a game, instead of explaining it with words.