A half shell (or alpine) helmet is the standard ski helmet you'll see on most mountain slopes. It covers the top and sides of your head and includes ear pads—either fixed or removable—for warmth and protection. It's by far the most popular helmet style for a reason: it strikes a great balance between impact protection, ventilation, and all-day comfort.
Half shell helmets are ideal for resort skiing and all-mountain riding. They pair easily with most goggles, offer enough insulation for cold lift rides, and still vent well when you're working hard. Removable ear pads give you flexibility to adjust warmth as conditions change throughout the day.
If you primarily ski or snowboard at resorts, this is almost certainly the style you want. It's really only the wrong choice if you're a competitive racer (who needs a full shell), a park rider who prefers a lower-profile freestyle lid, or a backcountry tourer who prioritizes ultra-light weight over warmth.