A circuit needs at least three electrical contacs.

motherboard circuit

A circuit breaker does not "cause" smoke. A circuit breaker "breaks" a circuit when there is too much current, creating a hazardous condition for the wires that are connected to the circuit breaker.

The circuit breaker PROTECTS you from electrical fire.

Find the source of the smoke; what burned?

If a circuit breaker tripped during the incident, it is usually caused by melting/burning wire insulation, either inside or outside of an appliance. If the insulation inside the walls of your house has burned/melted, it could be that the circuit breaker was too large for the wire or that the circuit breaker failed to shut off at the appropriate current load.

If the circuit breaker failed, your insurance should help you.

If an appliance overloaded the circuit, your insurance should help you.

If someone connected an oversized circuit breaker, causing the wire to overheat, your insurance company may refuse to help you.

Circuit breakers prevent excessive current from flowing in the circuit by "tripping" when the current exceeds the circuit breaker rating. This causes the current to be cut off, and is used to protect the wiring as well as the appliances.

Circuit continuity can be tested by using an ohm meter.