The Beatles are extremely famous for adapting the typical I-IV-V-I rockabilly progressions that were so popular before the 60s to new progressions that included augmented sixth harmonies as well as Neapolitan harmonies. This is evident in the song "Do You Want to Know a Secret" from the album Please Please Me released in 1963.
This song possesses one of the most "minor" sounding introduction of any Beatles song, and does so because of the presence of the Neapolitan (bII) chord introduced in the last part of the 3rd measure of the song. John Lennon sings "You'll never know how much I really love you, you'll never know how much I really care" in what sounds almost like a pop recitative. The presence of the Neapolitan chord on the second "really" adds a different flavor to the introduction of the song.
As far as its effect on the performer, the Neapolitan chords adds an almost "bluesy" feeling and perhaps begs to be emphasized. This is evident when John Lennon sings "really" the second time--it seems almost as if he is scooping down into it to emphasize its sadness. The lyrics and tune of this introduction serves as a way for the singer to express his concern with that fact that his lover does not understand how much he cares. Therefore, the Neapolitan chord provides the tension that is clearly delineated by the text. For John's guitar, this chord may also need to be stretched out a little to be emphasized. John does precisely that, and though it is subtle, the emphasis can be heard.
As far as the effect on the listener, it is one of perhaps surprise and knowledge of something a little "different." If one were used to the typical chord progressions of rock at the time, perhaps this would provide a new flavor to the music that listeners were not previously aware of. This chord also serves, as stated before, to emphasize the longing that is clearly present in the words. Soon the piece becomes more upbeat when the singer decides to tell his lover "his secret," but at the beginning, the listener is surely aware of the singer's longing.
Here's a link to a cute little George/John picture melange (sorry Ringo and Paul) that is accompanied by the song. Enjoy!