The easy answer to this question is that Jay Gatsby is the one who tried to recapture his past romance with Daisy, but the answer to this is not so simple.
One of the most famous quotes from the book comes in Chapter 6 when it seems as if Jay is trying to recapture his relationship with Daisy. Nick tells Gatsby that he cannot repeat his past and Jay responds, "Can't repeat the past?...Of course you can!" But this line isn't so simple. In his narration, Nick doesn't say Gatsby wanted to recapture his love for Daisy, but "he wanted to recover something, some part of himself, that went into loving Daisy." Then he says that since his days with Daisy, Gatsby's life has been "confused" and "disordered" and that he wants to "return to a certain starting place." Nick wants to make sure the reader understands that reclaiming the love with Daisy isn't the No. 1 priority for Jay, but making himself feel the way he felt before the war is what mattered.
This might seem like nitpicking, but the distinction between recovering his love for Daisy and recovering a part of himself when he was with Daisy is an important one. Before the war, Gatsby found out anything was possible, even a house like Daisy's. It was this idea of hope and dreams that Gatsby wanted to recover. Of course, this is not to say that Gatsby didn't love Daisy. He did and he hoped he could be reunited with her until the moment he died.
Tom and Daisy also wanted to recover parts of their past. Daisy wanted to feel the way she did with Jay before the war, which is why she engaged in the affair. But Tom also wanted to recapture his past. In Chapter 1, about Tom, Nick says, he "felt Tom would drift on forever seeking, a little wistfully, for the dramatic turbulence of some irrecoverable football game.
One might even be willing to argue that Nick's story is an attempt by him to perhaps recover something about Gatsby or perhaps to find a way to save Gatsby after his death.
The Great Gatsby is about how the world moves forward even when those living in it aren't ready for it to move forward. Or, in the words of Nick in regards to Gatsby's dream, "He did not know it was already behind him, somewhere back in the vast obscurity beyond the city, where the dark fields of the republic rolled on under the night."