Look at the bare wire the breaker will be protecting. If the wire is silver in color, it is aluminum. If the wire is brass in color, it is copper.
Look at the wire's insulation for its printed gauge. This is the diameter of the conductor and typically looks like 12/2 or 14/3. The first number is the gauge, the second number is the amount of conductors in the wire.
Choose your breaker size using the type and gauge with the chart below:
Use a 15-amp breaker for copper 14-gauge or aluminum 12-gauge wires.
Use a 20-amp breaker for copper 12-gauge or aluminum 10-gauge wires.
Use a 30-amp breaker for copper 10-gauge or aluminum 8-gauge wires.
Use a 40-amp breaker for copper 8-gauge or aluminum 6-gauge wires.