This is a common question and what you pack definitely varies from person to person.
The easiest way to break this complicated question down is to divide it into two categories. These are the staples:
Off the mountain; while most days will be spent skiing or boarding the mountain, the evenings will often entail a walk outside on cold slippery streets to go to a bar or find some food.
A jacket (preferably with a hood) is a must and if you have the money a down jacket is perfect. Many people will just wear their ski/snowboard jacket but you do risk having to wear it a bit sweaty/damp depending on how the day went.
Shoes/Boots are pretty important. You will no doubt see people walking around in Converse All Stars but I guarantee you they are cold and have slipped over at some stage. If you have the money, find a boot that has good grip and is pretty water resistant.
Gloves that protect your fingers from the wind chill are handy (get it) to keep in your jacket pocket and are pretty cheap.
Beanies and warm socks are self explanatory.
Layers are the key to keeping warm and the beauty of this method is it doesn’t cost much and can be adjusted according to the weather. If it gets really cold I would go for a thermal top and bottom, t-shirt, jeans, hoodie, and jacket. If it gets too hot you can just take of a layer.
On the mountain;
This list is assuming you own or borrow these items so if you plan on hiring you can ignore what doesn’t apply to you.
Snowboard or Skis
Bindings. Don’t forget to pack the bolts!
Boots. It might be an idea to try these on before heading off. New boots will benefit from being warn around the house for a few hours and old boots can change shape in storage so do yourself a favor and spend some time in them. Your feet will thank you later.
Snowboard / Ski specific socks. I normally pack 3-4 pairs. They can get expensive but having cold wet feet will ruin your day in an instant.
Thermal top and bottoms. I take 3 sets with me and they are a cheapish brand. You don’t need to spend big on thermals in my opinion.
Snowboard / Ski jacket and pants. There are a few things to consider when selecting your jacket and pants. Fit (not too tight and not too baggy), pockets (handy for your trinkets), waterproof level and breath-ability (basically the higher the rating the better). Look (lets face it. The mountain is an outdoor catwalk).
Face mask. I think it’s a must. If you don’t have something to cover your nose and below on a cold windy day, your nose and below might fall off. I prefer to go for a cheap lycra balaclava (found online for below $10). These tend to breathe well so your goggles don’t fog as much.
Gloves / Mitts. If you tend to feel the cold I would 100% go for a mitt over a glove. You do give up a little bit of dexterity but mitts are easier to get on and off and you can make a fist to warm your fingers. The other thing to consider when buying gloves or mitts in the material. Leather with Gortex treatment is the best and worth the money if you can afford it.