When choosing a tire, there are two main features to consider. Weight and traction.
Weight: Having a light weight tire means you can potentially go faster on the climbs, since you are literally carrying less weight up the trail. So why not go for the lightest tire? Because light weight usually means smaller knobs and less traction. This may still sound enticing, less rolling resistance also means you can go faster. However, less rolling resistance is meaningless if you are losing traction on the technical climbs and switchbacks.
Traction: In general, the more traction you have, the heavier the tire. That may not sound enticing, however, the more traction you have, the more confidence you'll have on the trickier sections of the trail. What gives a tire traction? The width of the tire as well as the spacing and size of the treads.
Most new mountain bike tire widths range from 2.2" to 2.6". The wider the tire, the more grip you have on the dirt. It lowers your rolling resistance a little but will increase your ability to stay connected to the dirt. The past couple of years, bikes have begun to accommodate wider tires because the width of the tire has such a profound impact on the quality of the ride. There is a limit for width though, the "fat' and "plus" tires may give you more grip, but less maneuverability in addition to being heavy. Most bike manufacturers are finding the sweet spot to be around 2.3" to 2.5".
What does all of this mean for you? It depends what type of bike you are riding and what your goals are on the trail. I am going to focus on the 2 general categories. In general, most of my clients fall into the first category, riding trail bikes and wanting to build confidence in their skills.
Trail bike: Your bike has at least 120 mm travel if not more, and you want to build your skills and create confidence on the trails. Rolling resistance and weight are less important than predictable traction (especially when braking and cornering) and sidewall durability. A good combination is the Maxxis DHF 2.5 in front and Aggressor 2.3 or Aggressor 2.5 in the rear. The DHF is super grippy in tight corners because of the square knobs that dig in and hold on. The Aggressor has a fast rolling middle section yet it still has enough tread on the sides to keep you connected. These tires are super reliable, very durable and tubeless ready. Another great tire for the conditions in Park City is the Maxxis High Roller II. The Women in the Mountains coaches all ride Vittoria Martello 2.35 in the front and rear of their Jamis Bikes. The Martello is perfect for staying connected in the loose dirt and over the rocks, yet light enough that the ride is still nimble.
Cross country bike: Your bike is between 100mm and 120 mm travel, you are a confident rider, looking to get faster and you might be racing your bike. You need to decide how much traction you are willing to give up when buying a lighter weight tire. A popular combination is the Maxxis Rekon Race /Maxxis Ardent Race, they balance being lighter weight and maintaining some traction to keep in connected in the corners.
These are just a few popular tires, there are lots of other combinations and brands of tires that work really well. Our coach in New Zealand has more tips for tire selection. Another consideration is a tire pressure, take a look at the short post I wrote to determine what tire pressure to run.