The Short Answer:
Arpeggio is an RPG system designed to have high flexibility but low mathematical complexity. Whether or not it achieves either of these goals is a subjective opinion.
The Sales Pitch:
Arpeggio is a fusion of Dungeons and Dragons* and Paper Mario that incorporates elements from Fire Emblem and other video games. Arpeggio is a system for creating role-playing games (RPGs): Arpeggio is not one game with a specific setting, it is instead a set of rules for how to handle the numbers of an RPG, and these rules can be used in multiple different settings. In a game of Arpeggio, an Arbiter presents a fictional world to one or, ideally, a group of players through narration and non-player character dialogue, and the players dictate the actions and dialogue of their player characters within this imaginary realm. Arpeggio is designed for use with a Mario-esque universe, wherein the player characters will face enemy characters in turn-based battles, but will also be challenged with field puzzles (which often involve platforming) and have the opportunity to interact verbally with non-player characters in the field. By solving field puzzles, the player characters will progress spatially through the imaginary world, and by defeating enemies and gaining experience points, the players will increase their fighting prowess. However, as was the case in the original Paper Mario battle system, base offense and defense increases do not occur through leveling up, only through special events in the plot of the game. Field stats will never change throughout the course of the game, but in character creation, each player has the opportunity to create a unique field ability for his or her character that will allow the character to accomplish things or reach locations that no other character can. Players design special attacks to be used by their characters in battle, and characters can equip weaponry to increase their offensive potential and armor to better defend against increased offensive potentials. A wide variety of Status Conditions, Weather Conditions, and other factors will hinder and/or aid characters both in battle and in the field, and players will have opportunities to create additional attacks for their characters through leveling up. Different options are provided for unarmed special attacks, armed special attacks, psychic attacks, and magical attacks. Like Paper Mario before it, Arpeggio is designed to minimize the mathematical complexity involved in both battle and field calculations, so that the Arbiter should rarely need to do more than flip a coin to determine the outcome of a situation or to perform simple subtraction to determine the damage of an attack. However, many features do factor into gameplay, including the building of support between teams of characters. Above all, Arpeggio is meant to be an enjoyable experience for both the Arbiter and the players. If at any time you are playing or regulating Arpeggio and you find yourself experiencing a negative emotion caused by this situation, you should immediately plead the 42nd and Only Amendment to the Rules of Arpeggio, which states the preceding purple sentence. If this fails to resolve the issue, then you should stop playing.
See also hegel5000's (albeit unfinished) article on RPG theory.
*Arpeggio is not directly based off of Dungeons and Dragons, but rather based off of Allo's Town, an instant messaging RPG which uses the Fuzion system, which is a ripoff of something called Cyberpunk 2020, which is (apparently) a ripoff of Dungeons and Dragons that attempted to go in the complete other direction, focusing more on role-playing than battling. Arpeggio is therefore about as related to Dungeons and Dragons as Lanky Kong is to Donkey Kong, but all of this is rather difficult to understand, and still harder to express succinctly, hence this footnote.