The story
Gaṇeśa is the remover of obstacles, and saṅkaṣṭa is the obstacle itself — the day is named for what it is kept against. The Gaṇeśa Purāṇa gives it to a king whose kingdom was restored after he kept it.
The moon is written into the day by an older quarrel. The moon laughed at Gaṇeśa when he stumbled, and was cursed to wane; the curse was softened so that he waxes again, and Gaṇeśa's day now ends by looking at him. What was mocked is what releases the fast.
When it falls on a Tuesday it is Aṅgārakī Caturthī, and is held to carry the weight of all the year's others at once — Maharashtra keeps it on a scale that empties offices early.