Poseidon, who supported Troy in the Trojan War, is angry at Odysseus for being a powerful Greek warrior who helped defeat his favored side. Odysseus is also a source of anger for being under the protection of Athena, an enemy of Poseidon's. But Odysseus especially raises the ire of Poseidon when he blinds Poseidon's son, Polyphemus.

Odysseus blinds Polyphemus because the cyclops has penned in he and his men as if they are livestock and has already begun eating them.

Initially, Odysseus has the wisdom to identify himself as "Nobody." However, as he and his men are sailing away from Polyphemus, Odysseus's pride gets the better of him. He wants Polyphemus to know exactly who bested him, so he cries out his real name.

Odysseus makes a grave mistake in identifying himself, because the cyclops is, as mentioned, Poseidon's son. Once Poseidon knows he hurt Polyphemus, his anger grows. This makes Odysseus's last part of his journey home to Ithaca particularly treacherous.

Odysseus made a lapse of judgement that not only threatened him but his men as well. However, being reminded of the importance of remaining anonymous in dangerous situations helps him when he returns home and must kill his wife's suitors.

In The Odyssey, Odysseus tells the Phaeacians the story of his adventures and misadventures as he journeys home to Ithaca from the Trojan War. He explains how he and his men landed on the Island of the Cyclopes and how they sought assistance there. Instead of offering them hospitality, the Cyclops Polyphemus actually eats six of Odysseus’s crew members. Odysseus devises a plan to blind the monster and then escape from his cave hanging from the belly of a ram, and the plan works. Earlier, Odysseus had told Polyphemus that his name was “Nobody,” so when Polyphemus cries out in pain and the other Cyclopes run to help him, he yells from inside his cave, “Nobody’s killing me now by fraud and not by force.” So the others tell him to pray to his father, Poseidon, if “nobody” is hurting him. It’s a cunning trick on Odysseus’s part.

However, as Odysseus and his men sail away from the island, he shouts back at the monster in anger and pride: “Cyclops— if any man on the face of the earth should ask you who blinded you, shamed you so—say Odysseus […], Laertes’ son who makes his home in Ithaca!” Thus, Odysseus reveals his true identity and thus makes himself a target of Poseidon, Polyphemus’s father. Polyphemus then prays, “Hear me— Poseidon […] — come, grant that Odysseus […] never reaches home. Or if he’s fated to see his people once again […], let him come home […] a broken man—all shipmates lost […] — and let him find a world of pain at home!” Poseidon hears this prayer and takes his revenge on Odysseus.

Odysseus made the huge mistake of blinding the Cyclops Polyphemus, son of the sea-god Poseidon. Polyphemus told Odysseus who his father was, but Odysseus didn't believe him. So not only did he put the giant's single eye out—he also taunted him cruelly as he left the island and foolishly revealed his identity for good measure. Polyphemus cries out to his father for revenge, and Poseidon obliges, regularly intervening to ensure that Odysseus' journey home is as long, hard, and as dangerous as possible.

Poseidon's animus towards Odysseus is also related to the fact that the other gods's decision to allow Odysseus to leave the island Ogygia—where he'd spent seven years with the gorgeous sea-nymph, Calypso—was made while the sea-god was accepting a sacrifice in Ethiopia. Feeling sidelined and disrespected by the other gods, Poseidon makes his anger felt in no uncertain terms. Not long after Odysseus sets sail from Ogygia, Poseidon whips up a violent storm that destroys the hero's ship and almost drags him to his death beneath the sea.

Odysseus first aroused the ire of Poseidon after he and his crew came to rest on an island inhabited by Polyphemus, a cyclops. Polyphemus was extremely displeased by the arrival of Odysseus and proceeded to eat some of the crew. Naturally, this upset Odysseus, and he and his remaining crew then tricked and blinded the cyclops in order to escape. Unfortunately, they did not know that Polyphemus was the son of Poseidon, the sea god. The cyclops called upon his father to wreak vengeance on Odysseus for maiming him.

Odysseus also revealed his name to the cyclops, which meant that Poseidon knew the identity of the man he wanted to target with his anger.

Poseidon may have felt generally unreceptive to Odysseus because of his attachment to Poseidon's rival, Athena, but it is the cyclops incident that really makes Poseidon angry with Odysseus personally. 

Poseidon was angry at Odysseus for three reasons. First, Poseidon supported the Trojans in the Trojan war and Odysseus was a member of the Greek forces that defeated the Trojans. Second, the goddess Athena was the main patron of Odysseus and she was a rival to Poseidon, having beaten him in the contest to be the patron deity of Athens. Third, and most important, the Cyclops Polyphemus was the son of Poseidon. In book 9 of the Odyssey, Odysseus and his men blind Polyphemus. Because Poseidon is god of the sea, offending him caused numerous delays in Odysseus’ sea voyage home.

Why does Poseidon hold a grudge against Odysseus even though the other gods favor him?

The problem with the Cyclops answer is that Odyssey's travels began long before that, and the detour to the Cyclops cave was already part of Poseidon's dislike of Odysseus

Why does Poseidon hold a grudge against Odysseus even though the other gods favor him?

There is also a very specific, personal reason why Poseidon develops a grudge against Odysseus. On the way back from Troy Odysseus and his men land on the island of the Cyclopes, fearsome one-eyed giants, and he ends up outwitting and blinding the Cyclops called Polyphemus. Polyphemus's father is Poseidon, and he prays to Poseidon to avenge him. Poseidon in that moment conceives a deadly hatred for Odysseus and sets out to thwart him as much as possible. This episode is related in Book 9 of the Odyssey, and also referred to in Book 1.

Why does Poseidon hold a grudge against Odysseus even though the other gods favor him?

The main reason was that Poseidon favored the Trojans against the Greeks in the Trojan War (the reason for that goes way back to the “beauty contest” between Aphrodite and Helen). When the Greeks won the war, Poseidon took out his frustration and rage (he was always tempestuous) on Ulysses, the “brains” behind the victory. Another complication was how the Greeks were delayed at Aulis until Iphigenia was sacrificed – another dispute among the gods. A good way to understand the whole Trojan was is to examine how the rivalries in heaven among the gods acted out on humans on earth – a primary “belief” in the Greek world.

Why does Poseidon despise Odyssueus?

Poseidon, Greek god of the sea, despises Odysseus because of what Odysseus did to his son, Polythemus, otherwise known as the cyclops because of his one eye. Odysseus and his crew land on an island and find a cave full of lifestock that they proceed to kill and eat, little knowing that Polythemus owned them. Polythemus returns and blocks the cave with a huge rock that cannot be moved by man and then eats two of them each day. Wily Odysseus organises a plan for them to escape. They turn a huge branch into a spear which they use to blind Polythemus, then they strap themselves to the underside of his cattle, so when he opens the cave to let his sheep out to graze they can escape and go back to the ship and flee. Odysseus, having escaped, reveals what he has done to Polythemus who becomes enraged and throws rocks at the ship. However, because he is blind they miss. Finally, Polythemus raises his hands to the air and begs his father Poseidon to curse Odysseus, asking him to not permit Odysseus to return to Ithaca. If he did get back however, he must return by himself, his crew having been killed, and in a stranger's ship. Thus begins the animosity between Poseidon and Odysseus.