As promised to kick off a short tour of ladies' competition shoes, here are my newest pair, shiny, clean, new, and ready for my last competition of the season.
These are ballroom, standard, or court shoes, depending on who you ask. They enclose the foot because they are meant for heel-lead heavy dances like waltz and foxtrot. Heel height is customizable in most brands, though heights around 2.5 inches are common. Shoes are typically made from satin, though leather is also available; the color is typically flesh or nude, though white and black are also available. Since these shoes are fabric, they can be dyed to match a costume.
Several styles of straps and toes are available. The trick is to find a combination that stays on your foot comfortably. My shoes are Supdance 1012s because of they have the integrated strap across the foot and the rounded toe. I feel my feet are wider than average across the balls, so I like the strap because it keeps my heel from popping out of the shoe (a problem for me with strapless varieties). I also like the rounded toe because then I don't feel like I have an extra several inches of shoe to deal with in lines like the throwaway-oversway. These are personal considerations, and every dancer will have her own reasons for her particular style of choice.
These shoes were designed primarily for the international standard dances, but many American smooth dancers wear them as well. There are also dedicated smooth shoes available, which combine a closed toe with open sides. These are meant to give some support for heel-leads while allowing more flexibility for lovely pointed toes.
At the other end of the spectrum, we have latin heels. I like to think the entire point of these shoes is to attach a sole and heel to the foot as invisibly as possible. Latin heels are open-toed and open-sided. They are meant to make pointing the toe easy while pushing the weight forward toward the ball of the foot, good things for latin and rhythm dancing. Like for standard shoes, several heel heights are available, though I would say that 3 inches is the average. Satin and leather shoes are available, primarily in tan and black.
There are many different toe strap styles available, and which one you pick is personal preference. I didn't care for ones with many tiny straps or those with knots, because the tiny straps would snap easily and knots dug into the top of the foot if I ever had the top of my foot in contact with the floor (it would happen in formation routines). The ankle strap can be worn either around the ankle or around the foot by adjusting the buckle's position on the strap.
If you have to pick just one pair of shoes for multiple styles, I personally think that latin heels are more flexible. It is harder to produce good latin dancing in ballroom shoes than to produce good ballroom dancing in latin shoes. However, your technique will be helped by wearing appropriate shoes, and if you are looking to improve your technical skills as much as possible, be ready to get yourself the appropriate footwear.