Short answers, pulled from the story.
Ganesha is one of the most widely worshipped deities in the Hindu pantheon and holds the title of Pratham Pujya, the god to be worshipped before any other deity. He is the son of Parvati and Shiva and is recognized by his elephant head, four arms, and protruding belly. He serves as the remover of obstacles, patron of arts and sciences, and deva of logic, intellect, and wisdom.
Hindu Puranic myths offer several explanations. The most common holds that Parvati created Ganesha from clay, Shiva beheaded him in a confrontation, and then replaced his original head with that of an elephant. Another story says Shiva created Ganesha through laughter and gave him the elephant head and protruding belly because he found the original form too alluring.
The earliest terracotta images of Ganesha date to the 1st century CE and were found at Ter, Pal, Verrapuram, and Chandraketugarh. His classic form with well-defined iconographic attributes appeared in the early 4th to 5th centuries CE. The historian Narain concluded there is no convincing evidence of Ganesha in ancient Brahmanic literature prior to the fifth century.
The mouse, or rat, began appearing as Ganesha's principal vehicle in sculpture of central and western India during the 7th century. Interpretations vary: some scholars read the mouse as a symbol of desire, while others see Ganesha's mastery over the rat, a destructive pest, as evidence of his role as Lord of Obstacles. Martin-Dubost also records the view that the rat, which penetrates secret places, symbolizes Ganesha's pervasive reach.
In 1893, Lokmanya Tilak transformed the annual Ganesha festival from private family celebrations into a grand public event in Maharashtra. He did so to bridge the gap between Brahmins and non-Brahmins and to build unity against British rule. Tilak was the first to install large public images of Ganesha in pavilions and established the practice of immersing all public images on the tenth day.
Ganesha is worshipped or venerated across Southeast Asia, China, Japan, Nepal, and beyond. In Thailand, the cult was formally promoted by King Vajiravudh of the Chakri dynasty, and Ganesha appears in the seal of the Fine Arts Department and at Silpakorn University. Indonesia is the only country to have featured Ganesha on its banknotes, the 20-thousand denomination issued between 1998 and 2008. In Japan he is known as Kangiten, with the cult first mentioned in 806.