Black Rock City is a pedestrian city and the playa is especially tough on feet. You will be spending a lot of time walking during your time there and good footwear is essential.
Note: This section is extremely important - foot discomfort can really hinder your time at the burn. If you take nothing else away from this entire guide, let it be this page.
Boots Vs Other Types
Generally boots are preferred as they reduce the impact of the playa surface on your feet. They also help keep the playa dust off your feet (some designs are better than others for this) which will help prevent playa foot. That being said, what's most important is to bring whatever footwear you'll be comfortable in for long periods.
You won't regret bringing high quality boots, but will likely regret bringing cheap boots. I own several top quality pairs of hiking boots, yet bought a cheap 40.00 pair at Wally Mart for the burn because I was concerned about the playa dust ruining my good boots. That was one of the worst mistakes of my life - within 24 hours I had blisters on both feet and ended up wearing my sneakers all week. By the time we left my feet had a constant soreness and burning which finally went away about 4 days later. I won't make that mistake again.
Sadly most of the really interesting boots I've seen are terribly uncomfortable. This is especially true of men's boots. In the case of footwear comfort should trump appearance every time. The only thing I can suggest here is to embellish those ugly boots in some way if it bothers you that much.
Generally a well fitting lace up boot will offer the best protection and support. Just keep in mind what you will go through putting them on and off if you pick knee high lace ups. Slip on boots won't keep the dust out as well or provide as much ankle support. Lace ups with zippers are a happy middle ground - the laces can be used to adjust the fit yet they are easy to put on or take off.
Note that lace ups will get tougher to lace as the week progresses - especially if the holes do not have metal eyelets. Cloth and leather eyelets are inferior on the playa.
Unless you are exceedingly optimistic (or stupid), start wearing your boots well in advance of the burn and make sure they are well broken in before you hit the playa.
As part of your break in time you may want to take several long hikes a week in the months before the event. For those who don't normally get much exercise this is essential. You will have a much better time at the burn if you are not constantly sore and exhausted.
I've learned (the hard way) that changing between different shoes every day or two helps my feet hurt a lot less. I generally switch off between boots, cheap sneakers (preferably with velcro closures), and moccasins. Rotating shoes regularly helps avoid excessive wear on any one area of your foot.
Three Or More
In 2012 my fairly new pair of hiking shoes disintegrated within 4 hours of arriving on the playa leaving me with nothing but heavy lace up boots for the next 12 days. By the time I was leaving I had calluses on my calluses and blisters atop of those - every step was an exercise in pain and my first stop was Fernley to buy a pair of moccasins.
Since then I bring at least four pairs of footwear - Boots, Sneakers, and Moccasins for the playa. If things get really bad I can always break out my travel shoes in an emergency. The shoes I wear to and from the burn are placed in a zip lock bag to keep them dust free during the burn, but it is reassuring to know they are there if needed.
I try to always bring a pair of slip on shoes - usually moccasins. These are great for lounging around camp or late night runs to the porto potties. Last year my box truck floor was quite splintery so it was handy being able to slip my feet right into shoes before getting out of bed.
There is something extremely luxurious about being able to put brand new socks on at the burn so many experienced burners bring at least one new pair for each day. Some like to change socks several times a day, but I find that overkill - your mileage may vary.