There are things you really should carry all year round, a good spare tyre, a jack and wheel changing tools (including the locking nut adapters if appropriate). A bottle of water for topping up coolant, the wash bottle and even for drinking. A first aid kit which are even more important in winter.
Check your tyres, worn tread or cracked side walls aren’t safe, replacement tyres may not be cheap but will give you better traction in poor driving conditions.
Check your lights work, it might be worth investing in a replacement bulb kit – but make sure what you buy works for your car, in particular there a several different types of headlight bulb available.
Ice scraper and de-icer, check your windscreen wipers are in good condition, if not replace them.
Torch and spare batteries - or a wind-up torch – I also carry a head torch which lets me use both hands and still see what I’m doing.
Jump start cables are really in the carry all year group, but you are far more likely to find your car won’t start in the very cold damp weather, often the easiest way is to jump start it from another car (see below for how to use these)
Reflective warning triangle and a Hi Vis vest or jacket, making sure you have the best chance of being seen could save your life.
Road atlas, like almost everybody else I rely on satnav most times. There are times when it doesn’t work (flat batteries being a common issue) a road atlas (and knowing how to use one) becomes invaluable once again.
Sunglasses - the glare off the snow can be dazzling and for most of the winter the sun is a lot lower in the sky as we drive to/from work or do the school run.
Mobile phone charger, sometimes you need to phone for help or advice or to let someone know your are running late (or not coming because the road conditions aren’t safe. Being able to use your phone is important, being able to charge it …
In particularly bad weather or long journeys you may wish to consider the following:
Food and a warm drink in a thermos flask.
Shovel (a comment on the previous version of this post reminded me that car mats can be used to provide traction when your wheels will only spin in icy/snowy or muddy conditions.)
If you only do lots of short journeys try to take the car on a longer run once a week, or at least run the engine for 30 minutes weekly.