There are at least two bands who have used the name The Normals:
1) First wave punk rock band from Jefferson, Louisiana that spanned 1977 to 1984. Members were: David Brewton (Guitar, Vocals), Charlie Hanson (Guitar, Vocals), Chris Luckette (Drums, Vocals) and Steve Walters (Bass). They have an anthology cd out entitled "Your Punk Heritage".
2) Named in honor of their hometown of Normal, Illinois, this five-piece alternative CCM band comprised Andy Osenga, Mark Lockett, Cason Cooley, BJ Aberle, and Mike Taquino, longtime friends who began playing music together while in grade school. The Normals' debut LP, Better Than This, appeared in 1998; Coming to Life followed two years later. The bands last release, A Place Where You Belong came in 2002.