Charles Smith from Rdr2's Den

You can attack another lion for one of three reasons - skill, experience and territory. If you are successful with a skill attack, you will gain some experience and a small increase in skill. If you are successful with a territory attack, you will receive some experience and your territory will expand by 1 space. If you are successful with an experience attack, you will receive a big bonus to regular experience gains from a fight.

A lion can only be attacked for territory once per day, and can only be attacked once every 2 hours.

You cannot attack a lion who is more than 2 levels below yourself. The defending lion will receive a defensive buff, so keep this in mind before attacking.