That title is a phrase I didn't start hearing until I became a Catholic and started reading older Catholic books. You can find a bit more about the phrase here, and that blogger links to a reflection on the idea here. As Jen points out, introverts may not be so tempted to go out and join a million committees and volunteer activities, but they may still find their way out of duty by writing and reading and even inordinate praying (not that prayer can be overdone, but retreating to one's closet to pray while one's baby is screaming with a dirty diaper might not be the fast road to pleasing God).
Every once in a while, I used to search online to find exactly WHAT were the duties of my state in life. But the answer is simple -- it is my husband and children, my parents, and more generally, the practice of a Christian married life. I think I was probably searching to find the minimum so I could check off "duties done for the day." I suppose I was also looking for a way to feel good about myself -- sort of a grading scale. I did that and that and that, which adds up to holiness! But it's not so easy, when St Augustin says quite clearly the Church's teaching that we don't "earn" God's help nor can we expect to please Him by trying to get by with a passing grade.
I have been thinking about that duty thing because, for one, I am reading Abandonment to Divine Providence by Father Caussade (which reminds me a bit of Father Dubay's books -- I think it is because he sometimes SEEMS to overstate his case, but further thinking makes me wonder if he isn't right). Fr Caussade ponders on the meaning of the "present moment" and especially the idea of pursuing our duties in the present moment and leaving the rest to God.
The other reason I am thinking about "duties" is because a lot of the reading I picked for Lent is about improving as a keeper of the house and home. All the books I picked are hard going for me. I'm not a detail person -- I like to dive into books and read them through in a sitting or two. These books are almost all handbooks and I have to study them and then apply what I learned. So it's work, not recreation, reading them, but I think it was a good idea to devote Lent to this kind of thing.