This troper is going to go ahead and summarize for you: almost every single song featured in Clannad will either make you go "d'aww'' or start sniffling. Or both.
While the opening is incredibly beautiful, the ending counts as a Crowning Moment of Heartwarming, given how...adorable it is (and that is a word I don't use lightly, given my utter lack of a soft spot for cute things). It even spawned a meme in which the dango family in the animation were replaced with Pikachu. See it here.
The song is basically symbolic of Tomoya's feelings about Nagisa over the entire series. In the first season, it's adorable and twee. In the second half of the second series, it is heartrendingly sad. (Especially when Tomoya sings it to Ushio.) And at the end, it comes back, with the whole family singing it together, as...
~After Story~'s OP song, Toki wo Kizamu Uta, counts as a Crowning Moment of Heartwarming. Directly translated, it means "The Song That Transcends Time," which, put in context, is perfectly symbolic of Tomoya's love for Nagisa.
The lyrics are absolutely heartrending when placed in context with the series:
"It was only you who I loved, It's a song I only sang with you, It's the time we passed together, I can't go on all alone."
This troper is appalled that several other pieces are not here.
Nagisa's Theme. Dango Daikazoku without its vocals but it has a beauty of its own.
Thanks to the scene when this song is played ( Ushio crying as she and her father finally make up after so many years of loneliness), I will never be able to listen to it with dry eyes ever again.
The Gaiden Game, Tomoyo After ~ it's a wonderful life also has an excellent soundtrack. Life is Like a Melody in particular, is just amazing.
Nagisa Warm Theme Piano is possibly one of the most heartwarming themes on Clannad. When Tomoya finally meets Nagisa and decides that meeting her was the best thing that could have happened in his life this music is the best. It depicts quite well the love of Tomoya toward Nagisa.
Nagisa ~ Saka no Shita Wakare (Nagisa: Parting at the bottom of the hill): At first it's played over Nagisa's death and Tomoya's subsequent vision where he left Nagisa rather than lead her to the moment where she met an untimely death, making it a Tear Jerker moment to start. It then becomes Crowning Music of Awesome when it's repeated after Tomoya's second chance and realization that meeting Nagisa was the best thing that happened to him. The music starts as Nagisa survives Ushio's childbirth, leading directly into Chiisana Tenohira above to highlight a wonderful ending. I know these "crowning" tropes are supposed to be the "one moment" in a series, but there are too many here to limit it to just one, sometimes even per episode!
Town, Flow of Time, People plays whenever something sudden or amazing is happening; which means it plays at many of the greatest moments. A worthy song for it, too. Sometimes specifically starts at around 1:45 to maximize the music's impact.
Don't forget the vocal version of the same song. It's just as great and moving.
Roaring Tides, a theme which, like Town, Flow of Time, People, plays during some of the important scenes.
"A New Life Reborn": it plays only twice in the series, but is honestly one of the most effective pieces due to the placement.
