The title, The Remains of the Day, sounds pretty cryptic at first. "Remains" is a word that implies stuff left over, residues. "Remains" is also another word for dead bodies, which: yeesh. In what way can a day have a residue? Or a dead body?
Well, when you're an aged butler who has devoted his life to the service of a Nazi: plenty of ways.
Stevens is at the dusk of his life. He's a moldy oldie. The high noon of his career at Darlington Hall, when it sparkled as a center of interwar diplomacy, has long passed. On his first (and perhaps only?) vacation, Stevens has a lot of time to mull over the "remains" of his life—the memories, but also the consequences of his life choices.
He makes a bunch of stops on his trip, and these stops give him an opportunity to think over both the immediate past of the day's adventures and the more distant past of his years as Lord Darlington's butler. He's also mourning Lord Darlington's death with a respect that Lord Darlington did not enjoy at the end of his controversial life. Lord Darlington is the human "remains" of Stevens's life. Gross… but also kind of poignant.
The most explicit reference to the title comes at the end of the novel. Stevens, sitting on the Weymouth pier and mulling over a stranger's advice, thinks:
Perhaps, then, there is something to his advice that I should cease looking back so much, that I should adopt a more positive outlook and try to make the best of what remains of my day. (8.84)
The novel ends with his resolution to be an awesome employee to his new American employer. This is the closest that Stevens comes to shouting "carpe diem!" He's going to make the best of what remains of his day(s) by doing what he does best—being a super-loyal and detail-obsessed butler.