The lights on the ceiling fan go on when the fan is on, even though there is a switch for separate operation of fan and light. What went wrong, how can we keep them work independently (but sometimes work together when that situation is needed)?

Is it a remote fan? Or does it have pullchains? Make and model of fan?

Is there is a separate wall switch for the fan and the light, the fan black goes to the supply black and the red or black goes to the fan light wire, usually blue. Sounds like someone connected the fan and light wire together. If you have a remote, as "ceilingfanrepair" asks, you only need 1 wire to both, the remote controls things.

Separate switch? Then check out the wiring. There might be a loose connection on either the ceiling fan or the lights to their respective switch.