Hi Danyel,
We "demand" when we are asking for something in a firm and resolute manner. When we demand, we are unlikely to accept "no" as an answer.
Example: After being stranded at the airport for two days due to heavy snow, many travellers are demanding for compensation.
We make a "claim" when we are declaring something because we think it is the truth or a fact. However, it may or may not be a fact or truth.
Example: The defendant claims that the plaintiff was the one who instigated a fight.
When we have a need for something, we "require" it. We call that thing a "requirement".
Example: We require more information about the flight status.
Hope this helps.
Cheers, Lance