This book was originally titled "Elinor and Marianne," but Sense and Sensibility is barely a leap from there – the traits included in the title describe these two main characters to a tee. Elinor embodies "sense" completely; she's practical, intellectual, and logical in all things. Her younger sister Marianne, however, is "sensibility" all the way.
Here, we have to remember that "sensible" didn't always mean what it means to us today. We generally think of "sensibility" as basically being practicality, but back in the day, it actually meant kind of the opposite. In Austen's time, "sensibility" was closer to what we'd call "sensitivity." Marianne is totally emotional, sensitive, and wrapped up in her feelings (especially when they're romantic ones), and thus is the incarnation of Austenian "sensibility." The challenge at the heart of the novel is for "sense" and "sensibility" to cooperate, and for the sisters to find a meeting point between reason and emotion.