"Into the Groove" (Madonna)

I don't care if you love Madonna or loathe her.  "Into the Groove" (1985) is the kind of song that gets any party, wedding, bar mitzvah, wake, etc. on its feet and dancing.  
The track has the curious distinction of being one of her most popular songs, even though it was never released as a single in its own right.  Apart from being featured in her film debut, Desperately Seeking Susan, the song was released only as the B-side of the "Angel" 12" single here in the U.S.  And as radio-ready as "Angel" was, it's not even in the same galaxy as the immediate, bass-heavy electro-pop of "Into the Groove"—a song that always felt like a (not so distant) cousin to Shannon's "Let the Music Play."  Main difference between the two songs: whereas the timid Shannon passively waits for her man to come back to her on the dance floor, Madonna goes and gets what she wants because she's tired of "dancing by herself."  (Make of that what you will.)
She wrote "Into the Groove" with producer Stephen Bray, an ex-boyfriend from her days at the University of Michigan.  Bray created the song's signature bassline, hook, and rhythm track, while Madonna penned the lyrics.  Supposedly, she wrote the bulk of the words, sitting on the fire escape of her walk-up on the Lower East Side.  She did kind of a rush job because she was more interested in getting a date with a guy who was flirting with her from his own balcony across the street than finishing the song.  As she admitted to Rolling Stone in 2009, she feels a little silly singing "In the Groove" today because she considers it one of her weaker compositions.
"I've never been a good judge of what [songs] are going to be huge or not."

"Justify My Love" (Madonna)

If you were alive in 1990, you probably recall the controversy surrounding the video for "Justify My Love."  If you were old enough to stay up late and/or young enough to give a damn, you also probably watched the unedited video when it made its US debut on the December 3, 1990, edition of ABC's Nightline.  
I remember watching the video as a pre-teen and thinking it was the silliest thing I'd ever seen.  Yes, there were glimpses of people's bangers n' mash, but it wasn't nearly as raunchy and titillating as the media made it out to be.  The content certainly didn't warrant it being banned from MTV.  It was sexual hyperbole—an elaborate joke leveled by Ms. Ciccone at the puritanical world.  (And she laughed all the way to the bank when the VHS release of the 5-minute video became the top-selling music video release of all time.)
But so much for all that.
It's a song that sounds like no other.  Those whispered, non-sung lyrics* over that relentless, trip-hop beat** still sound fresh today.  And even though the song set the tone for what was on the horizon with 1992's Erotica, she never got quite this funky or seductive again.

* The song's co-writer/producer/background vocalist, Lenny Kravitz, was having an affair with Prince protégé, Ingrid Chavez, back in 1990.  Chavez wrote Kravitz a private love letter/poem, and that poem became the basis of the lyrics.  However, Kravitz used the words without crediting her.  She sued.  Yadda yadda yadda, they settled out of court, and now Chavez is listed in the song credits.
** The beat was sampled from the Public Enemy track "Security of the First World" without the group's permission.  It almost led to another lawsuit against Kravitz by Public Enemy producer Hank Shocklee.  But then questions arose about whether Shocklee had himself lifted the beat from James Brown's "Funky Drummer," so Kravitz escaped lawsuit #2.