They usually wield Swords along with Axes. Stat-wise, they are usually described as half way between a General and a Swordmaster. They don't have as much defense as Generals but more speed and they don't have as much speed as Swordmasters but better defense. Heroes are typically set to be one of the "balanced" classes of the Fire Emblem series alongside the Soldier class. However, many characters that are/can become Heroes typically become very powerful units.
A promotion to the Hero class is one that few can take, and it is the only one that does not occur automatically. Both Josh and Ben automatically become Heroes after Chapter 16, due to plot events. Other than those two, there are only 3 other heroes in the game. First, there are the bosses of chapter 18, SudoJosh and QuasiBen, and the boss of Chapter 28x, who is the Heart of the champion.
The only two characters that can become Heroes are essentially the lords of this game. As said above, both Josh and Ben become Heroes after Chapter 16.