An “arcade mode” is one sort of play mode available when you play a port at home, and as the name would imply, it is usually the only play mode when you play in an arcade.
In general, you run through the game’s story, going through successive fights, maybe being treated to cutscenes, and facing bosses at certain turns. If the game is a port from an arcade game, the arcade mode is intended to play very much like what you would play at the arcade (but home ports often add some perks that you might not find in the arcade).
For most games’ arcade modes, you can set the difficulty level on the A.I. in the option mode, making it possible to cater to beginners and advanced players playing alone.
Just like with the real arcade version, you can receive challenges from another player.
Some games call this same mode along the lines of “original mode”, “simple”, “play”, and so forth.