Red, brown, and green algae are not very closely related, but often have similar growth patterns and all can be useful as decor and for biofiltration in marine tanks.
Among the reds, I have had red grape (Botryocladia) in my tanks for years. It grows well, as long as it's properly anchored, and looks quite nice. People also use another red, Gracilaria, for refugia. In my experience, herbivores will eat the reds before anything else, and they make great tang food.
Although one does not see them as often, Sargassum species are probably the most widely kept of the brown macroalgae. When it does well, it can become a bit of a pest.
Of the greens, you'll see mostly Caulerpa, Chaetomoropha, and Ulva species in peoples' tanks. Some really like Caulerpa, but I have seen it move from fuges to tanks and become a pest. Chaetomorpha, aka spaghetti or brillo pad algae, does very well floating in refugia, and is easy to harvest. Ulva, or sea lettuce, will sometimes ride in on live rock, but in my hands it only does well when nitrate and phosphate are high.
Many of these species will arrive on live rock, or you can by them from places like
Inland Aquatics,
IPSF, or
Florida Pets.