Of all the seasons, Winter is without a doubt the most difficult to survive in. In most cases, the snow and ice will hide most of your resources as well as make every part of survival extremely difficult. Even in the mildest of conditions, survival in the winter is miserable at best but one of the most important types of survival to learn and prepare for if you are a hunter or if you spend any time in the wilderness. The single most dangerous factor of winter survival is Hypothermia. "Hypo" means under, (hence a hypodermic syringe delivers liquid under the dermis - the skin). Normal human core body temperature is 37°C (98.6°F), clinically hypothermia occurs when the core temperature falls below 35°C (95°F) - that's not much of a drop in temperature.
If someone is exposed to cold and inadequately protected, their body will first try to generate more heat through shivering to maintain a normal temperature. If the body can't stay warm by these means, it will start trying to decrease heat loss by decreasing blood flow to the extremities to minimize cooling. Finally, if the loss of heat carries on despite these measures, the body will slow its metabolism to minimize its need for fresh blood flow and oxygen supply.
If it is inevitable, then the sooner the body reaches the final step, the better chance of survival so that the organs won't become starved of oxygen. For example, a core temperature of 20°C (68°F) requires only 20% of the oxygen required at normal body temperature of 37°C (98.6°F).
Given these facts, your number one priority in a winter survival situation should be fire, followed by shelter. With those two items, you stand the best chance of surviving long enough to be rescued. A person can live weeks without food and in most cases you will have some form of water near by, so the majority of your time should be spent on firewood and shelter. Only then should you spend time on the other needs.
Survival in cold weather is drastically different from warm weather. Your priorities, as well as your equipment will be different from what you are used to when dealing with warm weather conditions. If you have followed the preparation steps, you are properly dressed, and you have a well thought-out survival kit, you should be able to survive at least a week and maybe longer.
About this image: This was taken in 2014 on a winter camping trip in Eastern Oregon. My dog Sydney when she was a puppy on my ATV pulling an Aluminum cargo trailer that I built a couple years earlier. It is totally water-tight and hauls everything we need for a weekend camping trip. It has independant torsion bar suspension on each side to damper vibration and carry the load (up to 500lbs). It has a rack on the back that will haul two 5-gallon gas cans and a 2.5 gallon propane tank on the front to supply gas to my lantern and cook stove. On top it has a rack that I built to haul light-weight items that don't need to be in the water-tight space. It also has a removable table that mounts on the front over the propane bottle to place the cook stove on.