Algae are photosynthetic, but they are also unicellular organisms (Euglena) and shares some of the common features with plants as well as animals. They move like animals and perform photosynthesis like plants. Therefore, they are classified into Protists, and not the Algae. Their most features resemble the non-photosynthetic protozoa, and not plants, and therefore, are put into the Protista.
They also lack a cell wall, which is a feature of plants.
Some scientists advocate their classification in plants because of their sexual mode of reproductiona, and formation of spores.