Seaweeds are simple marine algae. There are thousands of seaweed species. The most common colors of seaweed are red (6000 species), brown (2000 species), and green (1200 species). There is only one known species--Lyngbya--that is toxic to humans.
Non-culinary uses of seaweed
Three classes of industrial gums--alginate, agar, and carrageenan--are extracted from seaweed. These polysaccharides are used because of their gelling properties and high viscosity. Seaweed industrial gums are a half-billion dollar market. Agar is widely used as a microbiological substrate. Alginates are used in the manufacturing of textiles and medical dressings. Carrageenan is present in topical lotions. All three classes are added to foods to modify their consistencies. Alginates can be found as stabilizers in ice cream and carrageenan shows up in many candy bars (I assume it adds chewiness).
Seaweeds are used as fertilizer. Potato farmers in Ireland and Scotland layer soil with seaweed. Organic farmers use coralline algae as a substitute for chemicals they deem unnatural. Liquid seaweed extracts are used in gardens, glasshouse crops, and citrus.
As you might expect, seaweed is reputed to cure health problems such as tuberculosis, arthritis, colds and influenza, high cholesterol, tumors, and worm infestations. While seaweed's medicinal uses are largely unsubstantiated, there is little doubt that seaweed, like most vegetables, is healthful.
Seaweed as a food
Culinary use of seaweed is most popular in China and Japan. In Japan, seaweed accounts for 10% of dietary consumption. The most popular seaweed varieties in Japan are kombu (kelp), nori, hijiki, and wakame. The Japanese use seaweed in soups, salads, sushi, and seasonings. In other parts of the world, dulse, alaria, and laver are common foods. Seaweed adds a salty, vegetal flavor to dishes. Sea vegetables contain a higher density of vitamins and minerals than any other food (or so says an online natural foods company).
Reference: http://seaweed.ucg.ie/defaultsunday.html
Responses
Gritchka comments that algae is not a real biological classification and that calling seaweed algae is misleading because that term is so vague. gn0sis points out that Nodularia, a seaweed, is poisonous and causes problems in Finland. Upon further research, I have found that some seaweed blooms release toxins that are damaging to marine life and commercial fish operations. | 
