Hi Gigi,
Require has the same meaning as 'need'. They can be used almost interchangeably. For example:
'You are required to buy me lunch' and 'You need to buy me lunch' both mean the same thing, but require sounds a little more forceful and even more formal. It doesn't sound very natural to say 'I require a hug', but 'I need a hug' is perfectly correct.
Demand, on the other hand, is like an order. If someone says:
'I demand that you buy me lunch'
There is the implication that you will be in trouble if you don't buy lunch. You don't have a choice. At a base level, it also means 'You need to buy me lunch', but 'demand' sounds much angrier and carries more authority.