A fine example of a sacrificial part in a machine. Shear pins are actually found in many different types of machines1, but this writeup uses just one of the most common examples to illustrate.
"Damn! I just broke the snowblower!"
...So goes a typical first experience with the shear pin.
You have not, however, ruined your snowblower. It's just that, shortly after your second time this month running over a newspaper, some sort of plant matter, or other heavy debris hidden under the snow, the machine has sacrificed its equivalent of the jesus bolt to avoid even greater damage. If you planned ahead, you have a replacement shear pin and tools to install it easily and quickly. If not, you put away your martyred machine and start cursing shoveling... and procure a new shear pin as soon as you can get to town for a replacement.
Where / What / Why
People living in snowy locations maintain their own small snowblowers if their circumstances make it more economical to handle their own snow removal work than to hire a commercial plowing service. The various snowblower designs share this common feature: an intentional weak point which, by failing in the event of excess torque in the machine, prevents damage to the motor and driveshaft. In most cases the shear pin is not actually a pin; it's a bolt — but the terminology abides. Perhaps it's less troublesome to think of the shearing of something known as a "pin", since our language gives bolts a noble position among the ranks of objects intended never to fail.
Sooner or later, when operating a snowblower, you'll encounter an unseen obstacle that the machine cannot pass, something that either catches and stops the blade, wedges itself into the works or winds itself about the driveshaft and stops the shaft from turning. Such stoppage would damage the motor or break the driveshaft if it persisted. For this reason, instead of a continuous driveshaft, these machines have an interrupted, two-part driveshaft, with the two segments reunited by a weak little bolt. This bolt — the shear pin — is either deliberately undersized for its job, or made intentionally out of an inferior alloy, or made with a weak point (a stress riser) in its structure, or some combination of the above.
Replacement
Sure, at the moment you're in a rage and would rather melt down the shear pin, murder its creator, gun down the paperboy who left the newspaper where it became snowed over on your driveway, burn the trees that dropped branches on your front walk, trample the kids who left small toys on your lot, and so forth. But look, you might as well replace your shear pin so you can get to work and earn another chance.
Go get yourself a source of light, an adjustable wrench or a pair of pliers, and a replacement shear pin (this includes both a bolt and a nut to fit). What, you don't have a spare? Only a fool lets himself run out of spare shear pins during the winter. You might as well just lie down in the snow and wait to die, you're worthless. Oh all right, there's the spare where you'd forgot you hid it away. Make sure the machine is turned off and cannot possibly turn on. Rotate the auger in the front of the snowblower by hand to locate the shear pin — it will be fairly easy to find if you watch which parts move together and which parts should but do not. This brings you to the location where the shear pin should be connecting one half of a driveshaft to the other half, and there you will find either an empty hole or the broken remains of the previous shear pin. They are made to break cleanly to avoid difficulty in replacement.
Now for the actual replacement. After removing any remains of the martyred pin, make sure the hole where the new one will go is aligned properly. This usually requires rotating the two (currently independent) halves of the driveshaft until their shear pin holes align. Then the bolt should pass right through. On many machines it needs to be passed through from one side and not from the other: you can tell which is right by the fact that when you've got the right direction, the head of the bolt will be held for you by some other bits of metal so that you won't need to hold the head of the bolt steady at the same time as you tighten the nut on the other end. Having aligned things properly and inserted the bolt in the right direction (or either direction if there appears to be no preference), fasten the nut on the threaded end of the bolt. Do not over-tighten the nut or you will make your new shear pin more likely to fail prematurely.
Single vs. multiple: Many common snowblower models have multiple shear pins — for example, a primary one in the main driveshaft, and two secondary ones that each drive only a portion of the auger. The point of this is that if you're lucky, an obstruction will shear only one of the secondary pins, which costs a bit less and is slightly easier to replace. It should be fairly easy to tell which is broken by observing which parts of the works are able to turn independently of each other. For this and many other reasons it is worth repeating that you must by all means ensure that the machine cannot possibly turn on while you are testing.
That's all. Having just read this, if you own or maintain a snowblower, please go and make sure that you have a number of spare shear pins in reserve. When you are trying to leave your home for an important occasion and you shear your last pin, and you have no replacements, and the hardware store is closed for the day, you're in one sad state of affairs. Don't say I didn't warn you.
1. anthropod pointed out that many machines make use of shear pins and that it's worthwhile clarifying that snowblowers provide just one common example of them. Good point!