Snow is an excellent insulator. Snow inherently has so much air inside of it that it traps up to 95% of heat transfer. Also, because of this very principle, the snow closer to the ground will be warmer than the snow at the surface. This is why snow caves can be so comfortable. I have dug many snow caves, and spent many nights 10 ft deep in a Pacific Northwest snowpack, and while snow caves are super warm, they are time consuming to create. With two people, the fastest I have been able to dig out a proper cave is about 2 hrs. These days, I pretty much always just dig what is known as an improvised snow shelter. The improvised snow shelter doesn't have quite the insulating ability of a proper cave, but can be dug in 30 minutes and, if sealed proper, can retain a lot of heat without the moisture associated with the dripping roof of a slept-in snow cave.
shelter (A. Stephen)
The hardest part about camping and sleeping directly on snow is staying dry and warm. The first step in this process begins when you leave the car. Throughout the day, including when you begin to dig out your shelter, be cognizant of keeping crucial layers dry. Any items that are impossible to keep dry such as gloves and socks, you should double up on. While down jackets and sleeping bags are warm and packable, consider bringing either a jacket or sleeping bag that is synthetically insulated, since these pieces will retain their warmth even if they get wet. If anything you are wearing starts to get full-on soaked, switch it out or find a way to dry it out immediately. Drying out clothing in your sleeping bag at night is possible, but keep in mind that anything that is completely soaked is highly unlikely to be dry by morning, and the more wet things you pack around you, the more potential there is to spend the night shivering instead of sleeping.
An emergency blanket will come in handy to line the floor of your shelter so as to provide a bit of reflective heat, as well as a barrier from the snow in case your bag slides off your pad in the night, or if you need to make a quick exit. If you are religious about staying dry, chances are you will be able to escape being cold, but going to sleep with a nalgene filled with boiling water can go a long ways toward making sure of this.