I have a Catholic prayer book which contains a nice prayer to St. Joseph. After the prayer it says, "Indulgence of seven years and seven quarantines each time that this prayer is said." What exactly does this mean?
Quarantine means 40 (the popular etymology is that when we section someone off and call it a quarantine, it comes from making sailors wait forty days before getting of the ship during the black plague)
So seven years and 280 days
The time refers to the ancient canonical penances. Basically, say you commit Y sin, then by canon law the penance was, e.g., 7 years of fasting, a certain number of years among the prostrators, etc. In some way or another these penances still existed until the 20th century  (and even now in the East). But no one in the East now, or in the West for a long time actually does such penances. Basically indulgences arose out of the practice of mitigating confessional penances. Go on a pilgrimage and the Church will supply for the 7 years fast that you normally have to do.
It doesn't take long to see that, in objective terms, the person is doing less penance and the Church is supplying it and not merely waiving it. Hence the days and years of an indulgence remained because of the connection to canonical penances, but also as a measure (this indulgence is worthy the penance of 7 years and 280 days of fasting on bread and water).
In the new norms, under Paul VI, a subjective measure is employed instead. Basically the remission of temporal punishment due in proportioned to the charity and merit of the act you perform
Person A says a whole Rosary, but with a tepid devotion- the Church supplements that with an amount equal to what she put in, so basically doubling the value as far as remission from temporal punishment is due
Person B says a whole Rosary, but with great devotion. As much as his devotion is greater than A's, so much greater is his partial indulgence
In the old system they would receive an equal indulgence.
But note, an indulgence is merely a remission of temporal punishment. A 7 years indulgence remits said amount of temporal dues, but makes me no holier for it. I do not myself merit any increase in charity/sanctifying grace by getting an indulgence. It is the good act I do considered in itself that would be meritorious. Hence even in the old system Person B is better off, because he advances in grace more.