Two possibilities, one is good news, one not so good.

The key is to look at the axle s you try to loosen the right nut. If it turns with the nut, which is fairly common, it's just a slightly stuck thread, and fairly to solve. But if the axle is stationary, as the nut turns, the thread is stripped, possibly from over-tightening in the past. How bad it is depends on whether it's the nut or axle which is stripped. There are various ways of freeing a nut stripped this way, the easiest of which is to use a so-called nutcracker


Or you can cut it off with a Dremel cut off disc, or even a hacksaw by cutting one side on a diagonal, and spreading it open. Once the nut is off you'll know where you stand.

OTOH - back to the good news. If the axle turns with the nut, you can usually remove it by tightening the left side and holding it tight while you try to loosen the right. Or buy a 10mm axle nut, cut through one side with a hacksaw, then thread it on and grab it with a vise-grip, such that the clamping action squeezes the nut onto the axle, and holds it while you try to spin the right side nut off.

Good luck either way, and hoping it's anything but a stripped axle.