Many good points above.
The good news is that you should learn most of this on the initial walk-around where the BM shows you how things are. With luck they will have it in writing.
But do have a list, and don’t hesitate to look at it and check off things during your first visit. The BM won’t think that’s rude, they will be happy that you are making sure to have your questions answered before you commit, only to later find something that is a deal-breaker for you. 
Storage for your stuff, and how it is arranged (space in the tack room? locker? trunk in front of the stall?) is very important and will have everything to do with how well your time in the barn will go. It isn’t until a few weeks or months in that this really comes home! Also, make sure to understand blanket storage, and think about all of the layers and weights of blankets you commonly use in your climate.
I also recommend asking about farrier and vet. Some barns schedule regular farrier/vet visits for routine care of all of the horses, others leave it to the owners to do it on their own. Some barns want all boarders to use their own farrier/vet. Those may routinely hold horses if the owner isn’t there. Others encourage specific farriers and vets, and will cooperatively help with those visits if the owner can’t be there. But in that case they usually will not help with an outside farrier or vet. And other barns never help with vet/farrier visits, and you’ll have to make arrangements accordingly.
Some barns will pay the farrier, even the vet, and then bill boarders for the cost. Others want a check ready from the owner.
Be aware that if a vet is doing routine vaccinations, floats, and minor injury care for all of the horses, and is billing the barn for all on one bill, you will probably not have a separate identifiable record for your horse. It may be only “vaccinations for 12 horses at Sunny Acres Farm”. Something to ask about if the barn is arranging all of this.
Look at the the feet of the horses in the barn carefully and see what you think you can live with, if the barn is arranging the farrier care. There are BM’s that are somewhat indifferent to good farriery, and of course others who are very particular.
Ask about worming protocol. Does the barn have all the horses on the same worming routine? Do they handle it themselves to make sure it is done effectively? Do they test before deciding on wormer and when to worm?
Someone mentioned fly control. Does the barn put all of the horses on a feed-through? My current board barn does and it is working well. It has less fly problems than anywhere else I’ve boarded.
Everything mentioned in this thread are things that sound very detailed at the beginning, but they are important details to live with in a board barn. Don’t assume that everything will somehow be peachy keen, do ask about everything you are curious about. Good BM/BO’s will be glad to answer and have a new boarder who is well informed up front. If the BM/BO sounds reluctant or uncomfortable to answer, move on to another barn.
