Bastet originally was a lion goddess; almost all representations we have of her dating to the Old Kingdom depict her with the head of a lioness. In this period and later she was also associated with other lion goddesses, such as Sekhmet (a goddess of Memphis), Hathor (a goddess of Dendera), Tefnut (a goddess of Heliopolis) or Wadjet (a goddess of Buto, actually a snake goddess and the uraeus of the king and gods, but quite often depicted as a lioness). Because a lion plays an important role in myths around the eye of the sun god Re, Bastet was also equated with this eye, and could be called the daughter of Re.
However, in later tradition Bastet became a cat goddess, and this is how she is most commonly known to us. The cat was, according to Greek authors, a sacred animal; many bronze statuettes of cats were dedicated to Bastet and mummified cats have been found in large numbers in places where Bastet was worshipped.
Bastet herself is usually depicted as a woman with the head of a cat; many amulets and bronze statuettes show her this way. Often she is portrayed carrying a sistrum, the instrument usually associated with Hathor; a text about this goddess in Dendera says that she is "raging like Sekhmet and friendly like Bastet", which not only describes her dual nature, both destructive and peaceful, but also, again, shows the link between the goddesses.
Bastet was at home in the Delta town Bubastis (the name of which is derived from the ancient Egyptian words for "sanctuary of Bastet" (per Bastet), which can still be recognised in the biblical name Pibeseth (Ezekiel 30:17) and the modern name Tell Basta). She was the mother of Mahes, a lion god, as well as of (in other traditions) Anubis or Nefertem.