Proper footwear is not only safer, but more comfortable and easier to ride in. Here are some examples of proper and improper footwear:
The boot on the left is too clunky and heavy; even though the heel at least one inch high, the rider's foot is likely to get jammed in the stirrup if he/she doesn't follow proper riding form. Also the heel is so high that this boot will make it more difficult for the rider to keep his/her heel in the proper position.
The athletic shoe (on the right) is also a poor choice for horseback riding. The lack of a heel makes this shoe unsafe for riding; the rider's foot is likely to slip around in the stirrup more than it should, and could also become jammed much too far into the stirrup.
The boot pictured in the middle is an excellent choice for riding. It is a paddock boot designed for riding, and it has an appropriate heel height, stiffer sole, and will give the rider's foot more support in the stirrup. While the rider's foot can still slide forward in the stirrup, the boot design will likely keep the foot from becoming stuck in the stirrup, which is important if the rider should fall from his/her horse.
It's not necessary to purchase boots that are specifically made for riding, but you should choose a boot or shoe with a similar heel/sole design, i.e., neither too clunky nor flat, and the heel should be 1 inch to 1 1/2 inches high. Work boots and boots used in rainy weather may be appropriate, but boots with aggressive treads are not ideal as they can hinder your ability to get out of the saddle quickly. Check your closets before running out and spending $$ to see if you already have an appropriate riding shoe or boot!
Not shown here are tall boots, which are also a great option for riding. Check out the mail order links below to see the variety of boot types that you can choose from, and how much they cost.