Power is a product of volts and amps. A circuit that will draw 1 ampere of current when 100 volts is applied will dissipate 100 watts. That is, 1 amp X 100 volts = 100 watts.
So, for example, your 1200 watt hairdryer will use 12 amps of, say for example, your 15 amp circuit. That leaves you with 3 amps, and is enough for a couple of lights in the same room, with little left over. So, for another example, if you add up all the Watts for your Christmas lights outside, and they add up to 1745, that's 17.45 amps and will, for example, trip the 15 amp breaker on the circuit.
Sometimes, it's not that simple to just upgrade a circuit breaker to a higher amperage. Before you think of doing this, check with a qualified electrician on safety and usage. My best to ya and hope this helps.