Duke Leto Atreides is the head of the Atreides family, lover to Lady Jessica, and father of Paul. He's also the guy everyone says is doomed from the start of the book, which is unfortunate, because we like him. He's brave, charming, and generous, and everybody just raves about how handsome he looks in profile. He's also perhaps the most underrated of all of Paul's teachers. We need to look at his role in the story if we want to understand how Paul becomes who and what he is.
Duke Leto's got a moral compass that works nine times out of ten. It doesn't matter if his own life is in danger, if others will go his way, or if he'll lose out in the end. There are several examples of this throughout the story. In Chapter 15, he rescues the spice miners at the risk of his own life and at the cost of a carryall. We have no idea how much a carryall is worth, but we're assuming somewhere between a lot and ungodly expensive.
Leto also does the right thing regardless of what tradition tells him to do. This is an especially important aspect of his character, because tradition can often make characters in Dune blind to new ways of doing things (see Stilgar and the Reverend Mother). For example, the Harkonnens have a custom that beggars may receive "water squeezing from the towels" (16.7) used to mop up the palace's floor. The Duke changes the custom and has his people give the beggars clean water until dinner ends. This may not seem like much, but remember that Leto is a duke with a lot on his mind. He didn't need to go out of his way to change the custom, but he did, anyway.
Paul's moral compass may not always point north, but he does adopt his father's attitude toward tradition. The Fremen tradition states that Paul should kill Stilgar to assume the role of leader. But Paul knows it's wrong to kill a man as useful as Stilgar. He changes the tradition, proclaiming, "it is time we stopped killing off our best men" (44.65). It's a trick he learned from his father, and it serves him well.
But even Duke Leto can't completely change himself. He moves to Arrakis on orders from the Emperor, but his heart remains with his home planet of Caladan. In fact, Leto thinks of Arrakis as "a hell [he's] reached before death" (11.14).
Leto tries to make the best of it, to inspire Paul. There are even times when he tries to see the good in Arrakis. While watching a sunset, Leto notices that the "night grew a faggot of luminous gray, then seashell opalescence that dimmed the stars" and thinks about how beautiful the scene is (13.73-74). But even here we see Leto's love for Caladan in the way he describes Arrakis in terms of a seashell's color—a subtle hint toward his longing for the oceans of Caladan.
Leto can't properly manage Arrakis, because he's never fully become a creature of Arrakis; he never assumes the "desert power." His failure ends in death, but Paul learns from his father's mistakes. He makes Arrakis his home, "gear[ing] [his mind] for its environment" (29.9), and this becomes the key to Paul's survival and eventual conquest of Arrakis.
A lot, actually. In Greek mythology, Leto is the daughter of the Titans Phoebe and Coeus. Yes, Leto is traditionally a girl's name, but the Duke wears it well, so it's cool. Anyway, Leto joins the long, long list of women whom catch Zeus' eye. Zeus, being Zeus, seduces her, and she gives birth to the gods Apollo and Artemis.
Atreides also comes from Greek mythology. It is the name given to the descendants of Atreus. Perhaps most famous of his children was Agamemnon, the man who commanded the Greeks during their conquest of Troy.
In the same vein, Duke Leto's child, Paul-Muad'Dib, combines features of both Greek legends. Like Apollo, Paul is a god to the Fremen. Like Agamemnon, he is a conqueror of countries. It's important, too, that both Apollo and Artemis (like Paul and Alia) can be symbols of both wisdom and power (in particular, mad skills with bows and arrows). So, Leto's name suggests the importance of his own legacy by hinting at its literary heritage.