One I think teachers find most interesting is the Wife's Prologue and Tale ("The Wife of Bathe"). However, I think this is not a student's favorite. I think there is nothing to relate to in her Prologue (be puzzled by or amused at, yes, but relate to, no). There is, however, something to relate to and a moral lesson to learn from her Tale, which is ironic since her tale is wise and she is so vulgar.
I think one that is often overlooked that should not be is the Knight's Prologue and Tale. His Tale is a true chivalric romance that is very engrossing and engaging. This Tale has appeal in terms of relating to students in that everyone can always relate to a romance, and students relate to the mystic of the chivalric era. In addition, mythological allusions grab interest in mythology; in the juxtaposition of mythology and Christianity in Chaucer's writing; in customs and social attitudes during the chivalric period.
A second one that is commonly used in non-American study of The Canterbury Tales is "The Nun's Priest's Tale." The tale, a best fable, is about that charming Chanticleer, who crows so well and is beguiled by the Fox because of vanity. Students will always relate to the concept of vanity tricking a person into courting doom or disaster. In addition, there is the interesting dynamic between Chanticleer and his favorite hen, Damoselle Partelote, who tells him that she "cannot love a coward" and that "ye lost my heart and all my love." In this tale, these two debate all manner of topic that was relevant to Chaucer's time and, in many regards, are still relevant today. Students will always relate to male/female dynamics and most likely willingly debate whether Partelote was justified and right or unjustified and wrong in her reaction to Chanticleer.