Swedish and German, a variant or diminutive of Katarina, the pan-European form of Katherine, from Greek "pure".
Popularised in the US, Karen and to a lesser extent Karin were introduced into the UK from about the middle of the 20th century. The spelling is very variable - a fullish list is given under Karen.
A correspondent [KC] reports that her name, Kareen is an Anglicized spelling of the German pronunciation of the name Karin.
It is not clear if Carina is truly a derivative.