Why flames of a fire rise even though gravity pulls things downwards?

Flames are hot and hence lighter than surrounding air and hence they rise.

Anka Software is correct. The difference in density between the gases in and around the flame and the air cause an upward force. There are many other upward forces -- someone jumps, airplanes fly because of an upward force on their wings. People shooting bullets in the air.

Gravity is a force. These other things are forces in an opposite direction.