Oskar's Dad was a jeweler who always tried to encourage his son to do something other than the jewelry business. Dad would tell Oskar he "was too smart for retail" (1.19) and was largely responsible for raising Oskar into the precocious child that he is by encouraging all his interests.
Dad is dead by the time the novel begins. He was visiting the World Trade Center on September 11; his store was actually somewhere else in Manhattan. He died in one of the towers, in the wrong place at a very wrong time. To make this even more tragic, he called and left his family multiple messages on the answering machine as he was waiting to die.
Dad's death fractures the Schell family big time. It's like a divorce, where Oskar feels like he has to choose between Dad's memory or his Mom. He's torn between the two, and says,
Every time I left our apartment to go searching for the lock, I became a little lighter, because I was getting closer to Dad. But I also became a little heavier, because I was getting farther from Mom. (3.41)
In flashbacks, we see Dad as an open-minded, fun, devoted, affectionate father. Oskar's Mom refers to him as being "definitive"—he's a rational guy who doesn't have much room for faith or mystery. But he tells Oskar thought-provoking fables about New York City's lost Sixth Borough. They read and cuddle every night before he tucks Oskar into bed. Oskar can't accept that he's gone.
When he finds a key in Dad's closet, the book's plot is set into a motion, ultimately ending with our realization that Oskar is overwhelmed with guilt because he heard Dad's last call but didn't pick up the phone. He knew what was happening and was paralyzed with fear. Even though the quest doesn't give Oskar any closure, it does help him appreciate the time he did have with his Dad.