One of the most important aspects of good horse care is plenty of fresh air, no matter what the season. Contrary to some opinion, it's not a good idea to lock up the barn doors and windows in the winter as it causes a decrease in air circulation. That's not to say that air should be blowing right onto your horse, but horses can withstand more changes in temperature far greater than we can.
Ammonia from horse urine is a major problem in many barns. Having fresh, circulating air can help keep the smell down. There are a wide variety of bedding options to use to help absorb the urine, even just a plain dirt floor. However, many horse owners use some sort of bedding to soak up the urine and keep down the smell. What you use depends on your preference and the kind of facility you have. Many race tracks still use straw for bedding. They pile layer upon layer and the urine seeps down to the bottom layer, keeping it away from the horse's respiratory system. Once every couple of weeks, the entire stall is stripped - a huge job - and soiled bedding disposed of.
Most stables use some sort of shavings - cedar or pine. Shavings are relatively inexpensive, provide some cushioning to the horse's legs, are readily available and easily composted as a way to recycle soiled bedding. However, there are some shavings you should never use - like walnut.
However, in my small stable, I use pelleted bedding, made for horses. They come in 40 lb. bags which are easy to store and transport. They absorb extremely well and last quite a while. Some of them have special additives that help with the urine smell, which is very helpful. I find them the best to use to make stall cleaning easy and they are quick to compost to eliminate the soiled bedding.
Not all pelleted bedding is the same - don't use pellets meant for fireplaces! Not only are they not absorbent, but they are treated to be flammable. My barn has many open doors and windows as it doesn't get too terribly cold in the Pacific Northwest in the winter so fresh air isn't a problem.
However, in the summer, I do notice that there is more of a urine smell when the heat intensifies. I use a product called PDZ, a stall "freshener" powder when needed. Because my open stall plan, religious stall cleaning and quality products, I have not had a concern with the amount of dust. I also have rubber stall mats under the bedding to provide the cushioning that the bedding doesn't provide.