The circuit breakers in my home are 120VAC, 15A or 20A breakers.
What would a surge protector (e.g. this or a power strip with surge protection) do that a breaker wouldn't do already?
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Surge protectors and circuit breakers both cut off power during exceptional conditions, but they have different purposes and react to different events:
Circuit breakers have only one job: to prevent the wiring inside your house from catching on fire. That's it, they do nothing else. They don't protect you from shocking yourself, or from lightning, or from your cat chewing through a power cord.
Circuit breakers work by monitoring the current (number of electrons) flowing through the wiring, and cutting off the power if the current is too high for too long (which puts the wiring in your house at risk of overheating). The higher the current, the quicker a breaker will trip: it may be just a split second or it might be hours. The idea is that a transient burst of power (e.g. when your fridge turns on) is not particularly dangerous.
The rating of a breaker depends only on the size and type of wire used in your house (which in turn is based on the expected usage). Typical sizes would be 15 or 20 amps for outlets and standard lighting, and higher, dedicated breakers for high-use appliances like stoves or hot water heaters.
Surge protectors have a very different goal: to quickly disconnect equipment from the power supply if there is a power surge. Surge protectors don't monitor the quantity of electricity, but rather it's voltage. An unexpectedly high voltage can force power into appliances beyond their specifications. Typically surge protectors cut off power when the voltage reaches 300V or 400V, which may be indicative of a lightning strike.
Surge protectors also have the job of absorbing energy due to a voltage spike.
Neither surge protectors nor circuit breakers do anything to regular minor fluctuations in voltage like a brownout.
In my opinion the need for surge protectors is somewhat overblown, since harmful power surges are relatively rare and there's no guarantee that a surge protector will be robust enough to prevent damage anyways. But they are convenient if you need extra outlets.
EDIT @Ecnerwal's answer is correct that technically surge protectors don't disconnect the power, they redirect (or absorb) it. And his answer has much more technical detail about the types and purposes of surge protectors.
Surge protectors do not "cut off or disconnect" power. They sit between the conductors and conduct when the voltage is excessive. Depending how excessive the voltage is and for how long, they may or may not survive any given event. At no point do they cut off voltage to a device, unless the device they are part of happens to fail in a manner that results in that, or for the Metal Oxide Varistor (MOV) types that fail short-circuit, the circuit-breaker upstream of them or fuse built into them may cut off power when they fail.
The rarity or frequency of harmful power surges will vary by locale. I use both "whole house" type Silicon Oxide Varistor (SOV) suppressors at the main panel (with surge capacitors as well, which respond slightly faster to very short voltage spikes) and plug-in-strip type suppressors (which are often MOVs) for computer/electronics equipment. I have an additional set of "whole-house-type" protection at the the top of my well, since it's a 110-foot run from the power panel and inconvenient/expensive if it gets zapped and I have to pull the pump up 300 feet. However, the best thing I've ever done for protecting computer equipment is to replace network wires that run outside between buildings with network optical fibers that run outside between buildings and don't conduct electricity.
SOVs have the claimed advantage of not drawing any current when the voltage is low, while MOVs draw a small amount of current at any voltage, and "age" as a result; likewise they are claimed to fail (when they fail) open-circuit, while MOVs are claimed to fail short-circut. Unfortunately (at least for independent verification) virtually all the SOV .vs. MOV information that I've ever found on the web is directly or indirectly from a single manufacturer of SOVs; on the plus side, a fair number of users have found the product effective in the real world. I'll refrain from mentioning the brand, as it will become obvious upon a search. Some other sources of information about (just) MOVs claim that they also fail "open-circuit," for instance (though that may mean "devices built with a fuse inline with the MOV" upon further reading - the MOV shorts & the fuse opens.)
"Spark Gap" or gas discharge tube suppressors should be mentioned here as well - they serve a similar function, but react much more slowly - sometimes they are combined with MOV/SOV types to increase the "size of surge" that can be dealt with, but pretty much nothing really survives a direct strike very well - direct strikes are best avoided.