By the looks of it, it's literally what it says on the tin - a straight, arcade-style marathon through the game.
The best comparison that immediately jumps to mind, is the Egg Shuttle Challenge Run in Sonic Colours - that took you on a straight ride through every single level in the game in order (except Game Land stages), with no cutscenes, and was focused on your score more than anything. You couldn't go back to other levels either, though you could use a temporary save or something IIRC.
In the context of Star Fox: Zero, this has more meaning though - in the Main Game of Star Fox: Zero, you don't get locked into the choices you make in terms of the paths, and some paths actually have to be unlocked by playing a later level first, and going back. Essentially, the main game doesn't really HAVE "paths" in the traditional sense - just a collection of levels, some of them having a special unlock requirement.
By the looks of it, Arcade Mode will be exactly like SF64 and SNES Sta Fox, in that you are locked into the choices you make, and don't have to unlock secret paths, as you have all of the necessary tools to get to them from the get go. I also assume you will have to start from the beginning if you get a Game Over, with your save data being erased if you get a Game Over. Oh, and it does seem that some of the side missions are not accessible here.