Horses (whether mounted or not) can either move or attack in the same round, never both.

Heavy Warhorses are the only mounts that can attack while mounted by a rider

Lances from a medium or heavy warhorse do double damage on a charge.

A rider gets a +1 to hit when attacking an opponent on foot that is armed with a weapon with a lower SR

Unmounted combatants suffer a -2 to hit when attacking a Fighter-classed opponent mounted on a horse.