I recently finished watching Clannad and the ending song is about dango and Nagisa, the one that wanted to start the drama club up again, always seems to be talking about or drawing dango. But, what exactly is a dango? We know it's a food of some sort because the main male character goes and buys them from a gas station and then Nagisa eats them, but what exactly makes a dango a dango and why is there a "big dango family" anyway?
A dango is "a Japanese dumpling and sweet made from mochiko" according to Shawn Balestracci's Wikipedia link. Mochiko is mochi flour, which is made from pounded rice. The Wikipedia page lists several varieties; some of these are mentioned in the song, such as the "peaceful an dango" source.
The Clannad anime is kind of unclear about where the song and characters come from, but within the world of Clannad, it seems to be some kind of mascot merchandising franchise similar to Hello Kitty. Nagisa has a dango daikazoku pillow, for instance. IIRC, Tomoya mentions early on that the dango daikazoku is old and unfashionable, but Nagisa herself is somewhat unfashionable, so it fits her.
Dango are somewhat commonly mentioned in anime and are even part of a saying, hana yori dango, which means "dango over flowers" (i.e., put food over aesthetic pleasures like flowers) and was punned on as the title of a shoujo manga. (The manga title uses different kanji to write "dango" that can also be read as "otokonoko" or "boy"). The sense I get from other anime is that dango are a common snack among Japanese young people, so they would be attractive as a mascot character for children and teenagers, leading to the idea of the Dango Daikazoku.
I think she mentioned it about it being a tv show she sayed it used to be popular, it's mentioned in episode 2, it's kinda after half of the episode, it's when they are making a poster to recruit members, and then he says the poster is missing something and he says she can put anything on the poster and then she asks if she could put her favorite picture and he said yes that's when she starts singing the Dango song after that then she starts talking about it.