There's no such thing as bad weather, only the wrong clothes. Keeping your hands warm in cold temperatures.
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Your hands have a large surface area compared to their volume, this means that they lose heat rapidly in cold conditions. In addition to this this, the muscles that operate your hands are mainly located in your forearm, you can feel this by holding one forearm with the other hand then and alternately making and relaxing a fist, the muscles can be felt tensing and relaxing. So even when your hands are used extensively, the heat generated does little to help warm up the hand.
This means that gloves or mittens are essential in cold conditions and indispensable unless you walk around all of the time with your hands in your pockets. Mittens will always be warmer than gloves if made out the same materials, but reduce dexterity. Gloves will never be as warm as mittens made in the same way.
As temperatures start to cool, the first call is to gloves that are thin and soft without a wind or waterproof outer layer. These will give you good dexterity, sensor gloves enable the operation of touchscreen devices. If you can put your hands in your pockets when not needed, these thin gloves will be sufficient for quite a while.
As the temperature continues to fall, heavier weight gloves with an outer layer that is wind proof and water resistant are the next step if you still need some manual dexterity, these are often described as ski gloves. If you need a greater degree of warmth then you need mittens, less expensive mittens will suffice if you don't need the dexterity.
In the very coldest temperatures a pair of insulated glove liners with substantial over mittens are needed. If you need to, you can slip off the mittens for finer handling using the gloves alone for a short time, with just mittens you would be bare handed without the mittens which could be uncomfortable, painful or even dangerous.
Good sized cuffs for gloves and mittens make them more effective as they cover the wrist and prevent a bare or thinly insulated region between the glove and coat sleeve, it all becomes one toasty insulated continuum. With thin gloves the cuff goes under the sleeve, with substantial gloves and mittens, the wide long cuff goes over the sleeve and has a draw cord or velcro closure to close it down, the most effective and warmest of all have cuffs that reach part way up the forearm.
Some gloves and mittens have a patch of fur on the back (often synthetic) that can be placed against cheeks or nose if they start to turn pasty white with frostnip.
For extra dexterity, fingerless gloves are available, I can't recommend them. They lack insulation at precisely the place where it is most important, avoid unless you know you like them. Some mittens come with a flip-top that exposes gloved fingers underneath if you need dexterity and there are even some that have a pouch where you can insert a chemical heat pack.
Recommended Winter Gloves
Note - giving gloves and mittens the name "Extreme" or labeling them "-20F!" or similar does not make them extreme or guarantee that they will keep you warm down at that temperature, so take it with a pinch of salt, it doesn't mean that they are not good though when used within their (probably lesser) limitations.
Most styles here are unisex or available in men's or women's sizes
Winter Gloves, OZERO - Deerskin Suede
Lightweight gloves, leather front, fleece backed
Black Diamond Cold Weather Mittens
High quality extreme cold weather mittens, men's and women's styles
Terramar Thermasilk Glove Liner
100% silk, warmth with dexterity or inner gloves in colder weather, in black or white
Burton Gore-Tex Mitts
Waterproof breathable mittens for serious conditions
Carhartt Men's Insulated Work Gloves
Tough work gloves for outdoor winter tasks
Outdoor Research Flurry, Sensor Gloves
For colder weather, water resistant, operate touchscreens without taking them off
with rechargeable battery packs, ski, motorcycle, arthritis
Alaska Bear Natural Silk Gloves
Lightweight naturally hypo-allergenic inner gloves, retain dexterity
Dakine Men's Scout Gloves
Ski/snowboard, waterproof insert, insulated, tough palms
Hestra Heli Ski Gloves or Mittens
Leather palms, seriously warm without excess bulk
Andorra Hyper Tech Touchscreen Mittens
w/ Pockets & Lightweight Inner Gloves
Dachstein 4 Ply 100% Austrian Boiled Wool Alpine
Hard wearing, tightly knit, the BEST wool mittens
Ojbro Swedish 100% Merino Wool (as Featured by the Raynauds Assn)
super soft, extra thick