Exigir: Claim something that belongs to you beforehand. For this reason: If a couple with children is separated, the mother would demand the maintenance of the children.
Pedir: Used to ask A favor, help, advice, a plate of food, a kiss? Haha
Demandar: Begin legal actions against someone ... although it can also be used when someone asks for "something" with great insistence ... a baby with his food is very demanding.
Mandar: Used to indicate the action of sending or, for the person who gives orders in front of something, a medical team, an army, the kitchen of a restaurant ..
Ordenar: Used for the action of placing an order in a restaurant, or to give me a precise indication / instruction of what you should do. It could also be the action of accommodating a room / desk.
hope it has been helpful! xx