As Ganesha is traveling through a forest in the night on his vahanam (mount) minjur, a snake scares the minjur and Ganesha falls off his mount.  He has just had his meals & his full tummy breaks open and all the sweets fall out.  The moon laughs at this sight.  The angry Ganesha curses the moon to lose his beauty.  When the moon begs forgiveness, Ganesha grants that he will wax & wane and appear in all his glory on the full moon day.

There are several versions on how Lord Ganesha got his elephant head.  The illustrations are for one version.
Siva & Parvati live happily on mount Kailasa.  Parvati is blessed with a beautiful son.  When all the Gods & Devas bless the new born on the cradle ceremony, Lord Saniswara avoids looking at the baby.  Parvati considers this an insult and asks Saniswara to bless the child.  Saniswaran is afraid that his glance will harm the baby, but on Parvati's insistence looks and blesses the baby.  The baby's head bursts into flames.  Lord Siva provides an elephant head to the baby and hence Lord Ganesha with his elephant head was born (head in the shape of the Tamil letter OM).  Thus Lord Saniswara's glance turned out to be favourable to Lord Ganesha.

Avvaiyar, Thriuvallauvar, king and the sangam poets
at the potramarai kulam