Snow shelters can be made in two basic ways. You can build the walls, or dig out the inside. These two methods can be accomplished in several different forms though.
Digging out the inside is good for places where there is a large abundance of snow, and it's easy to move, or where you have a natural feature (Large drift/bank) that you can hollow out. If you choose to make your own pile, make it at as tall as you, or a little taller, and let it sit and condense/harden for at least half a day if at all possible. When digging out, there are a few rules of thumb in thickness, but the easiest way is to poke a bunch of small sticks in the surface to a uniform depth (At least 8 - 12 inches) and dig out until you reach the butts of the sticks. Dig up and to the center of your pile, and then down and out, to make the stresses on the walls as small as necessary. Large snowbanks needn't be stuck with sticks if you are certain they have enough space for you to dig in.
You can also build a 'normal' shelter, basically a single man tent with whatever supplies you have (Don't use plastic, use breathable material) and cover it with snow. This is the easiest way to build a snow shelter when there is little snow around, or it may get warm enough to melt your snow. If possible, create a plug by filling up some fabric with snow, and shaking violently for a few seconds, then conforming to your door. This will harden, and allow you to cover the shelter with snow and not have to dig out your hole. If scraped all the way down to the ground before setting up, this shelter can easily warm _itself_ to 15 - 20 degrees above air temp, and more with you inside it.
In hardpack snow conditions, the easiest shelter is a trench. Dig down a few feet, pulling the snow out in blocks (Use a snow-saw if possible, or any other long, thin tool you can cut with). Lay these blocks flat over the top, or angled like a roof. If dug down deeper, you can go sideways to make sleeping benches and fit more than one person.
Igloos are difficult to make, and require practice ahead of time. That said, they work geat for mid-term camping in hard pack snow conditions. I hope never to camp for that kind of length in those conditions.
**Disclaimer** I've never made a snow trench or igloo. I've never camped in conditions requiring it. I hope never too, it implies cold and wind, and I've heard horror stories of windstorms eating away an entire igloo in a night. I have slept in an uncovered trench in snow, merely dug down to the ground, and I have slept in the covered shelter, and dugout shelter. Unless my bag is not rated for the outside conditions, I prefer simply scraping down to the earth, and sleeping under the stars.
Stryver