The swastika (Sanskrit "good luck" or "well-being", literally "blessed form") is a cross with its arms bent 90 degrees to either right or left. (Geometrically, it might be regarded as an irregular icosagon, a 20-sided polygon.)
The swastika appears in art and design throughout human history, symbolising many different things; such as luck, Brahma, the Hindu concept of samsara, or Surya (the sun).
The swastika is used primarily as a symbol by Hindus -- it was first mentioned in the Vedas, the holy texts of Hinduism -- but transferred to followers of other Indic religions like Buddhists and Jains.
A possibly spurious modern tradition has it that only a right-facing or clockwise swastika (as depicted above) is a good luck symbol, whereas a left-facing or counterclockwise swastika is a bad omen, labelled a sauwastika. There is little evidence of this distinction in Hindu history from which it is derived and, although the "standard" form is the right-facing swastika, Hindus all over India and Nepal still use the symbol in both orientations for the sake of balance. Buddhists almost always use the left-facing swastika.
In the early twentieth century the National Socialist German Workers Party adopted the swastika as its emblem and since World War II, most Westerners see it as solely a fascist symbol, leading to unfortunate assumptions about its pre-Nazi use.