This does indeed sound like a corn. Essentially a corn is a thickened area of skin on the foot in response to chronic pressure or irritation of the skin. When the skin is irritated, its natural response is to thicken as a protective measure. Corns usually occur on the outsides of the toes where they rub against the shoes, but they can also occur between the toes.
The first line of treatment for corns is reducing the pressure on the feet. This absolutely requires wearing only comfortable shoes that put no pressure on the feet at all. Additionally, you can buy small donuts and gels that can be applied around the corn to further alleviate pressure. Soaking the corn in the bath and then gently rubbing it with a pumice stone will gradually reduce the thickness and take the pressure off as well.
If these measures don't work, then you should see a podiatrist
. They can either cut the corn off altogether, or if there is another underlying cause of the corn (such as a hammertoe) they can suggest treatments for that as well. They can also help determine what kind of shoes would be least likely to aggravate the situation.