Let's start with this: Tire chains aren't legal everywhere. That would be the first "When Not" to use tire chains. Before you even set out to buy some, check if you're eveen allowed to put them on your car.
Tire chains are great in snowy weather, but pavement will destroy them, and some roads can experience damage from the chains. They could also destroy your tires or your wheels, brakes, wheel wells, and undercarriage. Make sure you can use them on your vehicle.
Now that we've gotten that out of the way, assuming it's legal for you to use them and your car can work with them, let's go over some rules:
The road(s) you're going to drive on with chains needs to have a completely compacted layer of snow or ice. There may be highways you can drive on that allow the use of snow chains, but if you hit patches of pavement, even if the highway can handle it, your chains probably won't.
Chains also add traction in muddy conditions. If you have to drive through, well, mud, then, by all means, use them. Keep in mind that your car will handle differently from how you're used to, which brings us to…
It shouldn't have to be said, but here goes. Don't drive normal highway speeds with chains on. Limit speed to about 30 miles per hour. Just because chains offer increased traction on snow/ice/mud doesn't mean that your car's going to be able to turn or stop on a dime.
Chains aren't magic. Slow it down. Now, you may be wondering if you have to put them on all four wheels. If your car has 4-wheel drive, yes, definitely. You don't have to put them on non-driven wheels (the front wheels of a rear-wheel-drive car, for example), but you may not have as much control as you'd like. Speaking of maintaining control…
Spend a lot of time adjusting the chains. If they're too loose, your tire will just spin inside the chain. Make sure you buy appropriate chains for your tire size and stop driving immediately if they become loose.
You may want to practice taking them on and off because unless you drive around with an extra set of tires (again, four if your car has all-wheel drive), you're going to be putting them on when you get to the snow/ice/mud you're going to be driving through. Of course, you may be able to avoid the headache of chains…
Use as Needed
Chains are a hassle. Yes, they're cheaper, but in many cases, you can get snow tires that will perform nearly as well in inclement conditions, and you don't have to worry about taking them on and off as the road transitions from snow and ice to dry asphalt.
Both options have their pluses and minuses. Tire chains may, in some cases, be required in certain conditions, so if you live in an area where that's the law, start practicing putting them on and taking them off. Just don't drive on any paved roads.
Using tire chains in snow and ice can maintain traction and keep you safe. There are only certain conditions when they should be used, hoewever. They must be installed properly, and checking whether they're legal in your area is important.