The Sketchbook is a band formed by the Sket Dan after their individual bands became unable to perform at the Kaimei Rock Festival. In the manga, they perform only one song - Funny Bunny by The Pillows, but in the anime, they perform an additional song, Michi.
The band won an award at the festival for "best song".
Though the Sket Dan compose the members of the band, the songs are performed in the anime by a real-life version of The Sketchbook, formed via The Sketchbook Project on the official website.
The band's name is a portmanteau of the names for the three defunct bands that the Sket Dan members were originally with: Sket Bombers, ch-ch and Yabasawa Books.
Sket Bombers + ch-ch + Yabasawa Books = Sketchbook.