Sehnsucht is a German literary term that is difficult to translate to English, or even to Western literary terminology. The most common translation is "yearning," and the word generally indicates a sense that something important in life is missing, along with a desire for other or new experiences. In The Great Gatsby, the two most important instances of sehnsucht are Nick and Gatsby.
Nick's sehnsucht is less overt; he is more of an observer, but he feels left-out of the high society East Egg culture. In the beginning of the novel, Nick is looking to become successful in bonds, and to become an accepted member of the high society; he achieves this through his friendship with Gatsby, and his relationship with Jordan Baker. However, his acceptance is superficial, and he ends the novel feeling even more alone and introspective. Nick's yearning was less for the glamour and more for the connection with people he admired, and he was heartbroken to find that those people had little integrity.
I spent my Saturday nights in New York because those gleaming, dazzling parties of his were with me so vividly that I could still hear the music and the laughter...
Gatsby's sehnsucht is more overt; his entire adult life has been set up to find and woo Daisy. His successes and the respect he gets from throwing money around are meaningless to him without her love.
"Can't repeat the past?" he cried incredulously. "Why of course you can!"
He looked around him wildly, as if the past were lurking here in the shadow of his house, just out of reach of his hand.
"I'm going to fix everything just the way it was before," he said, nodding determinedly. "She'll see."
(Fitzgerald, The Great Gatsby, mrbye.com)
For Gatsby, sehnsucht is apparent in his desire to return his connection to Daisy through material means. He has trouble connecting on an emotional level, and so he tries to recreate the past, yearning for a nostalgic time which may have never existed. He can shape the world in a physical sense, but he cannot erase the present and Daisy's emotional abuse at Tom's hands. Gatsby wants desperately for Daisy to love him for himself, but doesn't know how to express that longing except through wealth. Because of this, he cannot be honest with himself or with her.