Thundersnow is a combination of thunder and snow. It's pretty self explanatory, but surprisingly rare. The odds are stacked against any thunderstorm in the winter, let alone one that has the potential to create lightning. The biggest contributor to a system coming together to produce such a rare event all comes down to the height or thickness of the storm.
In the summertime a tall thunderstorm will be able to separate positive and negative charges in its cloud to create lightning. For us to have a thundersnow event we will need the same "charge separation." Only problem is that the cold, nearly freezing temperatures make it very difficult for a winter thunderstorm to grow or thicken-up. Accordingly, where there is no lightning there is no thunder and no thunder means no thundersnow. On average, a typical snowstorm will only rise to 8,000 ft where's the summertime cousin will push more than 35,000 ft into the atmosphere. This is why lightning and thunder are more frequent in the summer and not winter.
It has been asked a number of times if lightning can strike during a snow storm. The answer is, yes. In fact, if you hear thunder during a snow storm then a lightning strike has already occurred. Granted it could have been a bolt going from one cloud to another, but a lightning strike from the cloud to the ground can't be completely ruled out.
Does Thundersnow = A lot of snow?
Yep! Unlike springtime thunder, the density of snow limits the ability of thunder to travel long distances. In fact, some estimates put the ability to hear thunder during a snow storm at only 3 miles or so. So if you can hear thunder during a snow storm you are at most 3 miles away from the initial strike. Since lightning strikes are more frequent in the strongest part of a storm the ability to hear thunder means you are pretty close to the strongest part of the storm. In turn, the strongest part of the storm usually produces the highest snowfall rates too.
Surprisingly, it is possible to have a thundersnow event that drops some hail. It becomes a meteorological trade-off however. The cold temperatures actually hinder hail growth, but the below freezing temperatures allow more hail to reach the ground rather than melt. I've never experienced this firsthand, but certainly sounds like something you would never forget.
Despite the comparison to severe weather in the spring, thundersnow tends to lack the true punch we often find with May through September severe thunderstorms. In the spring, we classify a severe thunderstorm as a storm capable of inch size hail, 58+ mph winds and/or a tornado. During winter months, it is possible to be under a a "severe thundersnow warning", but this is usually because of the strong wind, blizzard type conditions associated with the storm. Large hail, while very rare, is possible with a thundersnow event, but a tornado will not be possible in a thundersnow event. The atmosphere is just too cold for anything to go "tornadic".