A Greek hero was someone who had much arete. Arete is pride, honor, and respect. Arete was gained by winning battles honorably, and showing respect to all, even the enemy. For example, as in The Iliad, if someone was killed in battle and the body was taken by the enemy, one owuld expect the enemy to give it back so that the family could honor the dead. Giving back the body would show both sides that you had honor and respect, or arete.
Achilles is a prime example of a hero. He was very proud, as described in The Iliad, and is respected by his allies. Achilles is a great warrior, and has won battles honorably many times. In The Iliad, when Agamemnon took Briseis from Achilles, he took it as a great personal insult to this arete. Achilles felt disrespected and dishonored because of this act.
This event was where The Iliad begins to describe Achilles' flaws. Achilles withdrew from the war because he was angry at Agamemnon. Then his friend died at the hands of Hector, and Achilles transfered all his anger from Agamemnon to Hector. He then killed Hector and disrespected him when he was dead. Achilles kept the body and mistreated it. Achilles' flaw was dishonoring Hector; hubris disrupted Achilles' arete. Hubris is the flaws, like disrespect and dishonor, that attack the arete and take some of it away. So because Achilles disrespected Hector's body, he, in effect, was harming how others respected him.
Ancient Greek heroes and heroes of today are similar in some ways. A hero today might be well known, respected and admired. Everyone knew and admired Achilles, and he was respected (arete). However, a hero today may be dedicated and hard working, which are not described as arete, and also patriotic. Ancient Greeks had no concept of patriotism. A Greek didn't fight for his country, he fought for personal glory.
Even with flaws, a hero was important in ancient Greece, and heroes of today are similar and also important.