What does the flame represent in Buddhism?

Answer 1

Originally it was a practical gift, not a symbolic one. And originally, people gave oil lamps rather than candles.

Giving oil lamps to the community of monks was meritorious because this provided light, thus enabling the monks to be active for more than 12 hours a day. So it was similar to other practical gifts, such as food, robes, shelter, medicine, etc.

Electric lighting is a relatively recent invention. By that time, the custom of putting lights (oil lamps or candles) on the shrine was well-established, and so was not discontinued. In poor countries, one sees that the oil lamps or candles on the shrine provide actual light in the evening and early morning.

Different schools of Buddhism may have developed their own explanations for what the lights represent, but these explanations were developed after the Buddha's passing.

Answer 2

Desire in all its forms.