circuit breaker trips out but power is still on

If I understand the question correctly - "circuit breaker trips out but power is still on" then the circuit breaker is not breaking the circuit and thus the circuit breaker needs urgent replacement by a quailified electrician.

This just mean the circuit that is supplying your power is not the same circuit that is tripping out. But you really need it checked out.

the circuit could be getting feed from two breakers. with a lot of properties you find this fault as work has been done to them through the years often tradesmen mix up and double fed a circuit.