It depends on the discipline. In western you should always wear smooth bottom boots so your foot can easily slide out of the stirrup in case of an emergency.
as long as they are strong boots
Boots or sneakers.
It depends on what time of year your cattle drive is in and where you are at. If it is warm and dry wear cowboy boots or ropers. If it is cold and wet or just cold wear insulated riding boots. I assume you are riding a horse as well.
Because they wont get caught on the saddle and they are just nicer to wear. It's so they won't get caught in your stirrup and they slide out easier and if you wear like joggers, since there is no tongue in horse riding boots it won't get caught in the stirrup.
The next best thing to horse riding specific boots are smooth soled work boots with a low heel. You could also wear 'fashion' style boots so long as they have a 1.5 to 2 inch heel on them. If you can only wear tennis or athletic shoes ask if the person has toe cages or Tapaderos to put on the stirrup irons to keep your foot from going through the iron.
shoes
The best kind of shoe to wear while riding is a boot with a small heel - about 1/2 to 3/4 inch. If you are riding bareback, or on a saddle without stirrups, You can wear tennis shoes, but the heels on riding boots help prevent your foot getting stuck in the stirrup, in the case of an accident. If you are a serious horseman/horsewoman, you should purchase some paddock boots, or perhaps field boots. For those who compete, the best options are dress boots, field boots, or paddock boots WITH chaps. However, there may be certain rules as to what you can wear.
Shoes!
Unless you're riding for show, or in full dress, it doesn't matter as much what kind of boots you wear. As long as the boots have a good heel on them so that your foot doesn't slip through, either paddock boots or tall boots can be worn. (Or any other pair of boots.)
You can wear casual clothes with a horse riding helmet.
If you mean riding a horse: DON'T DO IT!!!!! If you wear an ipod while riding, you could not hear noises of things that could spook your horse, or comments from your instructor. You could be severely injured. You would put yourself, the horse, and the people around you at risk. Save your ipod for working out or riding in a car(not driving).
usually a boot that is made for riding, bought from your local tack store. I ride in paddock boots and half-chaps, but other's ride in full length rubber or leather riding boots if you ride english. For western you where cowboy boots. Or for either you can save money and just wear plain old rubber boots.