नीला आसमाँ सो गया (female)

This is the Lata Mangeshkar version that sings about lost love.

"The night got killed when the moon disappeared in [my] tears.
Loneliness is what fills in [the rest of my] life"

The music punctuates the narration of the story nicely to bring out the emotion. You can relish the male version here.

आदमी जो कहता है

My blogger friend Atul has a more detailed discussion of this song. Briefly, this is a simple song that packs philosophy in every day words and experience.

Whatever a man says and hears [in his life],
Those voices keep following him throughout his entire life
Whatever [blessings] a man gives or is given,
[Those blessings] stay with him for his entire life

नीला आसमाँ सो गया (male)

It is rather rare that an actor or an actress also provides the singing based on a classical raag. Amitabh Bachchan does this well here. A musical pun, in a female voice later in the story, recounts the heartbreak.

दो लफ़्ज़ों की है, दिल की कहानी

"The tale of [two] hearts, is made of [only] two words ... love or youth."

If you interpret love or youth to mean true love or infatuation, you can see some justification in this simplified interpretation. It is said that Asha Bhonsle herself regards this song as one of her best renditions.

छूकर मेरे मन को, किया तूने क्या इशारा

"By touching my heart [or mind], what a gesture have you made?
This season changed, and the entire world seems lovely, wonderful."

Considering that an immensely desirable quality in a human being is to touch another at an emotional or at an intellectual level, it is only appropriate to have this quality immortalized in a song, crafted to hummable melody that only a Kishore Kumar can create.

मैं यहाँ तू वहाँ

A rather moving rendering of separation of two lovers.

कहता है बाबुल

I don't remember the circumstances of this song, but I do recall that a sincere-sounding voice sang this song in the background as if the dad's character in the movie meant it.

कभी कभी मेरे दिल में ...

There is a very valuable translation, and summary, of this song on the web and most of what is presented here is based on that: http://indiansongtranslations.blogspot.com/2008/07/english-translation-of-kabhi-kabhi-mere.html.