Chain battles are done against Noise. To begin a chain battle, the player must first gain a book on how to do so. From there when scanning an area with Neku, then pull a Noise towards him, and depending on how long the player wants the chain to be, drag more Noise to Neku within the time allotted before the match commences. Once the first battle starts and ends, there is no review phase, instead, just another match (and another for each additional Noise), once all the matches are completed then you may see your results.
This method allows for more experience and getting better as a player, since the Noise get harder with each successive battle, as well as more pins dropped and personal achievements.