DEMAND (требовать как необходимое; настоятельно спрашивать): I demand an apology (an explanation, an aswer); he demanded money from them; she demanded payment of the debt; he demands obedience; this task demands patience (great care, skills, energy); she demanded that he leave immediately; I demand to be shown the whole report; the suspect demands that he be allowed to call his lawyer; "Who told you that?" she demanded; demand angrily (indignantly, furiously, aggressively, irritably, coldly).
CLAIM (требовать как принадлежащее по праву; заявлять): he claimed the throne; she claimed her right to the estate; he claimed payment (compensation, damages); he claimed an exemption for dependents; he tried to claim back some of his expenses; this problem claims attention; she claimed against them / on them; he claims to be the sole heir; he claims to own the house; she claims that she was tricked into selling her house; claim rightfully; claim falsely.
REQUIRE (требовать как необходимое, установленное правилами): do as the law requires; you are required to register; I did everything that was required of me; the law requires us to pay taxes; the teacher required that her students should attend the conference; the officers required absolute obedience from/of their soldiers; she requires medical care; this task requires patience (great care, skills, energy); these flowers require a lot of water; this noun requires the definite article; the house requires painting; as circumstances may require; if required; when required.
NEED (нуждаться в; требоваться): we need bread, butter, and milk; we need to buy some meat; she needs help; I need your help; I really need you; I need a map of Moscow; call me if you need something; I need more time; I need to work harder; I don't need it; you don't need to do it; she needs medical care; these flowers need a lot of water; the water tap needs fixing; the house needs painting; this room needs cleaning; his shirts need ironing; if needed; when needed.