This is a large group of diverse unicellular and multicellular aquatic plants; they grow in both fresh water and seawater and are used commercially as a source of thickeners (agar, agarose, algin, carrageenan) and pigments such as beta carotene.
Some countries harvest algae as food. Spirulina is produced by Israel and Mexico; the Scots, Irish, and Canadians produce dulse; and the Japanese (and, increasingly, Americans) consume some species of Chlorella as well as seaweed like nori.
Some species of algae are important commercial pests because they clog pipes and foul pools, reservoirs and waterways (rivers are particularly susceptible to algal overgrowth that can kill fish when fertilizer is dumped in the water).
Specific algal divisions include:
Most of the information in this writeup was found in Introductory Plant Biology by Kinglsey R. Stern. The rest is based on work I did for the science dictionary at http://biotech.icmb.utexas.edu/