This is a really good question. Wearing proper athletic shoes will make or break your exercise experience.
When I first started teaching zumba I wore my same old athletic shoes that I was wearing for everything, from running the treadmills to walking outside and they were pretty well-broken in. After a couple of weeks my feet and knees started to really hurt! I thought it may be due to the shoes I was wearing so I went out and bought some Nike Air Max 360 shoes. They are classified as running shoes but for me all the air cushioning was exactly what I needed. I tried on different shoes and these shoes made my feet feel so supported so I gave them a try. As soon as I switched to these shoes my feet and knees didn't hurt at all anymore.
First of all, Zumba is not a dance technique class. If anything, it is an aerobic class so you need to wear aerobic type shoes. It is recommended to have shoes that have both lateral and vertical support as you would find in a cross-trainer. You also want to look for a shoe that does not have too much traction on the bottom of the sole. Too much traction will make your feet stick to the floor during those gliding salsa moves! Usually running shoes have this kind of traction. My first Zumba trainer told me that when you do Zumba you should pick up your feet. That way the traction on the shoes won't get in the way.
That was awhile ago. Now with an increased popularity in fitness dance and Zumba, the shoe manufacturers have come out with a great assortment of shoes geared to the "fitness-dance" enthusiast. Some good ones are Nike Shox Electro, Ryka Studio D, Ryka Intensity XT3, and Nike Musique III SL. These shoes have a pivot point on the ball of the foot to aid in easy pivots and gliding for dance moves. The Nike Shox Electro have more cushioning than the others. Some zumba students are wearing the dance sneakers, such as Bloch Boost DRT Mesh Sneaker. These have a split sole for easy pointing of the feet and also have the pivot point. In my opinion these do not have the support you need for the "fitness" part of Zumba.
Inserts will make your shoes last longer. Take out the inserts that come with the shoe and put in a better quality, well-cushioning insert. Then replace the inserts every few months. This will extend the life of the shoe itself and make your shoes feel brand-new again every few months. I just bought a pair of inserts at Lady Foot Locker. They said they will replace them for free after a few months of wear up to a year!
Whatever shoes you decide to get, get the ones that suit you best. It's best to try on different shoes in the store. Choose between cross-trainers, dance-fitness shoes, dance sneakers or even running shoes. Look for shocks and cushioning, not too much traction on the sole, and a feeling of lightness in the shoe. I would not buy the cheapest pair. Your feet and legs are important! Your feet will thank you for buying the best quality shoe after doing zumba on a regular basis!
To get the best results from a shoe you may want to replace your shoes about every 6 months depending on how often you use them. Some instructors have 2 or 3 different pairs and switch them off between different classes. This somehow makes the feet feel better . It is also a little trick that dancers do, switching between different heel heights and different types of shoes during the course of a night of dancing!
I hope this helps you decide on a good pair of shoes. Have fun with Zumba and be good to your feet!