I don't think they're called accidental chords. I'm not sure though. I'll explain what accidentals are though....or even better, quote wikipedia:
An accidental is a musical notation symbol used to raise or lower the pitch of a note from that indicated by the key signature.
pretty much boiling down to using notes from outside the key to enhance the music.
Why would we want to?
Probably because music tends to get boring when it's limited to 7 notes (tends to only). When you're adding the other notes, you give yourself a better shot at originality.
Okay, that was a bogus reason but it is a point anyway.
The biggest reason we use accidentals is to be able to give our song a movement....and to create tension which can be resolved by homing in on the root of the chord which follows. It's like....light always seems to be brighter after you've been in the darkness for a while. Similarly, the prettiness in the sound always seems prettier after a bit of dissonance.
How it makes a song beautiful?
Well, you've got me there. I just know these concepts....I haven't mastered them. I don't think I ever will either. Still trying though.
Some concepts which use accidentals...You'll find these on this and almost any other music site:
a) Flat 5 substitution
b) Circle of 5ths
c) Diminished chords/scale as connector between chords
e) Chord structures (2,5,1)
See if this helps. I doubt that I know enough to be able to answer your queries but shoot them here anyway.
As for examples, now isn't a great time but I can try and make something up.
And surely you’ll buy your pint cup !
and surely I’ll buy mine !
And we'll take a cup o’ kindness yet,
for auld lang syne.