Circuit breakers, like fusible links (fuses), are rated according to the maximum amount of current (load) they can support before the circuit is interrupted. Many appliances have their own circuit breakers, typically rated at something less than 10 amps.
In general, breakers are designed to open (turn off the circuit) when the current has been slightly exceeded for more than a few seconds, or greatly exceeded for less than a second. If you're attached to the circuit, you'd be electrocuted before the breaker blew, but at least the house wouldn't burn down.
Home circuit breakers come in many sizes and configurations, ranging from 15 amps to 400 amps (for main disconnect), with intermediate sizes including 20, 30, 40, 50, 60, 70, 100, 150 and 200 amps, among others.
Breakers are also classed by how many poles they have, whether both poles open simultaneously, what additional safety features there are (GFCI, arc-fault), who makes them, and what model panelboard they are to be installed into.