Algae need carbon, water and nutrients to grow, same as us humans. The main nutrients necessary are nitrogen and phosphorus, they are called macro-nutrients. Other substances are only required in very small amounts, they are called micro-nutrients and include for example iron, copper, magnesium, zinc, and vitamins.
Algae can grow in nearly any medium that contains enough of those nutrients. These can be culture media mixed together in the lab, media that are naturally available, e.g. river water, lake water, sea water, or media made from waste products, e.g. waste water, landfill leachate, worm and compost tea (extracted from composted food/plant scraps). Some of them are very rich in nutrients and need to be diluted.