You need troops to attack other Lords. You can send attack marches with just troops, but including a Hero as their Commander will increase your troops' abilities and allow the Hero to Face-off against the enemy Hero. Tap the Castle of a Lord that is not in your Alliance and use the attack function. Select the Hero and troops to deploy in the Set Troops screen, then deploy them to have the troops depart from your Castle. Once they arrive at the other Lord's Castle, the battle will begin. If there are any troops alive after the battle, they will automatically return to your Castle.
Tip: Each Troop type has strengths and weaknesses against other types. Troops deal additional damage when attacking types they are strong against. Infantry deals additional damage to Cavalry. Cavalry deals additional damage to Archers. Archers deal additional damage to Infantry. Siege weapons neither deal nor take additional damage from other Troop types, but deal additional damage to traps.