alm is on the right track, but it's a bit beyond that.
The real difference is that an LED is a diode whereas a lightbulb is effectively a resistor.
So if you have a lightbulb on 120V AC power, it's the internal resistance of the lightbulb that is controlling how much current is drawn from the source. In the case of a 100W lightbulb, it's likely about 144 Ohms (P=V^2/R so 100W = 120V^2 / X...X = 144). The manufacturer of the lightbulbs design them to draw the correct current from the sockets and to optimize light output for the current drawn (and light temperature and longevity and a bunch of other things that aren't relevant here). The circuit breakers you mention are only in place to LIMIT the total amount of current allowed through to a certain device, they have no impact on the device itself. So in the above example, if you had 500 100W light bulbs all hooked up in parallel, each asking for .8333 A (the amount the bulb wants based on 120V supplies and 144 ohm internal resistance), the circuit breaker would see too much current flowing through it and shut down to protect your circuit (and probably your house wiring from burning down).
The differences with a Light Emitting Diode (LED) is beyond a certain voltage drop across the device, it will sink as much current as it can. See
http://en.wikipedia.org/wiki/Light-emitting_diode#Technology for a few more details about it. So say you hook up a bench voltage source across an LED that is set to 1V drop. If the voltage source can supply 10A, it will supply AS MUCH as it can past the biasing point. So what to do? Well, if you have a 1V supply and the LED is going to "eat" .7V of that in order to be biased on, you have .3 V remaining to play with. If you put a resistor in series with the LED, the .3V across that resistor will determine the current for the entire circuit! Boom, easy current limit. So if your LED can only hand 30 mA (max), you pop a 10 ohm resistor in there and your LED won't go "pop". Of course you can also do what alm said and instead supply your LED with a current source, but a voltage source + a resistor in series effectively IS a current source.