The number today, "'Till Tomorrow," isn't especially a character song or plot song, a very slight touch of both, yet it's probably the most important number in the show. That's because of the story behind it.
When the show's legendary director and co-writer George Abbott was looking for who to write the score for the show, he was intrigued by the young team of Sheldon Harnick and Jerry Bock, based on having seen the musical they'd recently done, The Body Beautiful. The thing is, though they each had a few separate credits, that one show was the only one they had ever done together. And it was not successful, only running a mere 60 performances. So, they were asked to audition for the job. They came up with a lovely waltz that they thought would evoke the era of the show, America around the time of World War I. That song was..."'Til Tomorrow." It got them hired, and the show went on to tie The Sound of Music for the Tony Award and win the Pulitzer Prize -- at the time, only the third musical to do so.
"'Til Tomorrow" ended up being used in the show. And has remained a standard ever since.
In the show, the young lawyer Fiorello LaGuardia helped a group of striking women at their factory, and in doing so also helped get their leader, Thea Almerigatti, out of jail after she'd been arrested for organizing the strike. By all accounts, Thea was an extremely beautiful woman, and Fiorello was infatuated with her. LaGuardia himself was short and overweight, though with a vibrant personality and heart. Shy and highly appreciative of the man who helped her, Thea agreed to go out with Fiorello, and a sort of relation of appreciation and admiration developed.
Additionally, after Fiorello got the local Republican Party to put him up for the U.S. Congress as a sacrificial lamb against the corrupt Tammany Hall, he shocked them all by campaigning so aggressively that he stunned them all by actually winning.
Before going to Congress, though, World War I broke out, and LaGuardia enlisted in the war -- promising the immigrant Thea that he'd help free her home village of Trieste, and telling her that he wants to marry her.
And so, before heading off to war, at a goodbye party, the two dance to "'Till Tomorrow."
Here is the rehearsal from the recent Encores! production of the show. With the song that got Harnick and Bock hired.