In most snow avalanches (excepting very wet slides), there is actually quite a lot of air in the snow. If you aren't killed by trauma, and aren't buried too deep (where the weight squeezes your chest so you can't inhale), then you can breathe for awhile.
What happens over time is that the CO2 you exhale builds up around your face in the snow. Also, the warmth of your breath tends to slightly melt the snow against your face, which then refreezes. This builds up an ice mask over your face, which tends to be impermeable to air. So over time you suffocate from some combination of excess CO2 and ice sealing your face. On body recoveries, the presence of an ice mask indicates that the person lived for at least awhile after being buried. That is why fast rescue is so critical. If you're not otherwise injured, and are dug out in less than roughly 15 minutes, your odds are fairly good. After 15 minutes, chances of survival drop very rapidly.
The theory of the avalung is that it has a mouth peice somewhat like scuba gear, and a couple of tubes and valves. When you inhale, it takes in air from one valve, and when you exhale it is vented out somewhere else, away from the intake. So you don't rebreath your own CO2 and the ice mask forms away from your face. Avalungs have been shown to work in some cases.
An AvaLung can buy you some time
IF:
1. You can manage to get the mouthpeice in while getting the "Maytag" wash inside the equivilent of a cement mixer.
2. You aren't killed outright by trauma (about 25% of avalanche deaths are thought to be due to trauma, not suffocation).
3. You aren't buried too deep (if buried deeper than about 2 meters almost no one survives, probably due to the weight of the snow).
See
AvaLung by Black Diamond, and also
AvaLung Review.
Not getting buried in the first place beats any peice of gear!