Why do we mount horses from the left? Is there any reason why we should or shouldn't mount from the right?

The tradition of mounting from the left dates back to the time when warriors carried swords on horseback. Since most people were and are right-handed, the sword was carried on the left hip to enable them to draw it quickly.

Horses were mounted from the left so that the right leg could swing up over their back without the sword getting in the way. Mounting from the right would be difficult, as the sword would get in the way of swinging the leg up, and might poke the horse.

So it really is nothing more than tradition. We mount from the left because we've been taught to. Mounting from the left resulted in tack that was designed to be buckled, cinched, or fastened from the left-- most halters, bridles, and saddles are left-sided.

It's actually a good idea to teach your horse to accept everything from the right side as well.
Xenophon-- an ancient Greek general who wrote the first book on horsemanship-- wrote that it was good to practice mounting from the right in case of an emergency. Today it's a good idea because it will teach your horse to be balanced and accept things from both sides.

It can also help to reduce strain on their shoulder. Xenophon didn't use stirrups (they hadn't been invented yet), but we do, and when we mount it pulls the saddle over the horse's side. Mounting from both sides equally can helps to balance things out and prevent the horse from getting sore. It'll probably help you to balance too-- notice that if you've always mounted from the left, and you try to mount from the right, it feels 'wrong' and is more difficult! If you do it enough you'll get better and it will become just as natural as mounting from the 'correct' side.

Of course, most horses will need to be accustomed to right-side handling if they've been handled from the left all their life. Most horses are one-sided to begin with, so working from both sides can benefit them by helping to balance them out.