I was lining up for passport control at CPH and saw some people providing their fingerprint. I wasn't asked to provide it. Is this is a one time thing or are fingerprints not required from US citizens? Thanks.
As a US citizen you were most probably entering without a visa. There wouldn't be a point in fingerprinting you at the border because they have no known good prints to compare them to anyway.
However, people who apply for Schengen visas are fingerprinted during the application process, and they may be asked to provide matching fingerprints when they enter the Schengen area, as additional verification that the person who enters is the one the visa was issued to.
(As far as I can understand the official motivation for this arrangement, the concern is that someone could send all their friends -- or paid stooges -- to attend biometrics appointments for visa applications in different names, and then if just one of them goes through they would travel under that identity. )