The single player mode can be played between two parallel story lines, the "Hero" story mode and the "Dark" story mode. The player has the option of advancing in either story at any time. The "Hero" story features Sonic, Knuckles, Tails, while the "Dark" story features Shadow, Rouge, and Dr. Eggman. Each dark character plays similarly to its corresponding hero character, respectively. Levels advance in the same order for both story lines.
Levels featuring Sonic or Shadow are regular platform levels, designed in much the same way as Sonic's levels in Sonic Adventure.The player's primary objective is to reach the end of the level. Levels with Tails or Dr. Eggman are based on shooting, allowing the player to control a mechanized robot walker and shoot their way to the end. The Knuckles and Rouge levels are based on hunting, with the characters scouring the levels for three keys.
A two-player mode is also available in Sonic Adventure 2, where the corresponding characters from the "Hero" and "Dark" stories are able to play against one another in racing levels, shooting levels, and treasure-hunting levels.
Source: Wikipedia, "Sonic Adventure 2", available under the CC-BY-SA License.
From the back of the box
There is a new threat to civilisation, even more powerful than the infamous Dr Eggmann, an immortal capable of overthrowing Sonic for good, his name... Shadow the Hedgehog. With the help of his dark allies, our Sonic look-alike is causing unimaginable destruction, has Sonic met his match? In the most thrilling Sonic adventure ever created, featuring a host of Nintendo Gamecube exclusive enhancements, you decide the outcome - choose to join the dark side and conquer the world or join Sonic, overcome this new evil reign and restore peace to the galaxy.