Algae are a non-monophyletic group, that is that it doesn't share a common ancestor, but share the characteristics of being able to photosynthetize, and the abscense of true vegetative tissues. They are more related to plants than animals, sharing the same type of nutrition (autotrophy) thanks to the acquisition of the chloroplast, which allows them to do photosynthesis. Actually, plants evolved from a group of Algae called Chlorophyta.
So, the correct classification of this organisms isn't neither as plants nor animals.