Vaaranamukha... is a song in praise of Lord Vigneswara (also known as Lord Ganesh, Lord Ganapathy, Pillayar etc), the Lord of all obstacles. Hindus, before starting any function, sings songs in praise of Lord Vigneswara to usher in his blessings for the smooth conduct without hurdles.
In Hindu mythology. Ganesh (Ganapathy) is the elder child of Lord Siva and Lordess Parvathi, the younger one Subramaniya. Mouse is Ganesh's vehicle for transport. Ganapathy is a very naughty, shrewd and quick-witted one. Once Subramaniya wanted to impress upon Ganapathy the superiority of his vehicle, peacock over Ganesh's mouse and challenged Ganapahy to a race around the world. While Subramaniya got on his vehicle Peacock and went around the world, Ganapathy went one round of his parents telling that they are the whole world as far as he is concerned, appeasing them in the process.
Legend says, Ganesh written down the great Hindu religious Epic - Mahabharata, dictated by Sage Vyasa, using his right tusk.
As to how he lost half of his right tusk, how he came to have an elephant head, there are many stories. Some of the stories behind these, and some more will be given in future updates.