This is a great question. It is probably best to speak of honor instead of respect, as honor covers our modern understanding of respect and more importantly the Greek cared about honor immensely.
Honor is the most important point of the whole book. When Agamemnon and Achilles each took a maiden as the spoils of war, Agamemnon, in time, had to give back his maiden as she was the daughter of the priest of Apollo. According to the story, Apollo sent a plague.
When this happened, Agamemnon took Achilles' beautiful maiden, Briseis. At this act, Achilles' honor was slighted. The wrath of Achilles blew up and he refused to fight. This caused a huge problem for the Greeks, because Achilles was the greatest warrior among the Greeks. To put it another way, the Greeks did not stand a chance to defeat the Trojans apart from Achilles.
From this perspective, we can say that the honor of one man nearly cost the defeat of all the Greeks.