Percy's father rules the sea. He is a member of the Big Three, the three sons of Kronos, along with Zeus and Hades. Just like the sea, Poseidon can be difficult to read and interpret – Percy doesn't know if his dad loves him or cares about him. But, over the course of the story, we realize that Poseidon is very proud to be Percy's dad.
For most of his young life, Percy was told that his father left town soon after Percy was conceived. Percy had never met his dad:
I don't have any memories of him, just this sort of warm glow, maybe the barest trace of his smile. My mom doesn't like to talk about him because it makes her sad. She has no pictures. (3.8)
This warm glow suggests that Poseidon has checked in on Percy every once in a while, even though his mom denies it. Sally Jackson tells Percy that his dad is gone for good, saying he's "lost at sea […] Not dead. Lost at sea" (3.10). Even though talking about Poseidon makes her sad, Sally tells Percy a little bit about him at Montauk beach:
"He was kind, Percy," she said. "Tall, handsome, and powerful. But gentle, too. You have his black hair, you know, and his green eyes." (3.92)
When Percy finally meets his dad in person on Mount Olympus, he finds a guy in a Hawaiian shirt and shorts – a very laid-back, surfer-dude, Jimmy Buffet-ish kind of appearance. And yet, Poseidon is a god, and so he is formal when talking to Percy. Percy tells us:
There was no clear sign of love or approval. Nothing to encourage me. It was like looking at the ocean: some days, you could tell what mood it was in. Most days, though, it was unreadable, mysterious. (21.55)
But at the last moment, Poseidon tells Percy, "whatever else you do, know that you are mine. You are a true son of the Sea God" (21.109).