Lord Ganesh is the elephant-headed God, who is known as Lord of Beginnings and the Lord of Obstacles. He is regarded as the son of Lord Shiva and Goddess Parvati. The image of the Lord has four hands, elephant's head and a big belly. His charioteer is a tiny mouse, which shows a unique combination of wisdom, intelligence and presence of mind. Lord Ganesh is worshipped before starting any new venture. (More information here and here.)
In the two images above, the deity's trunk points to the right, symbolic of renunciation of worldly things.
On this old calendar image, the trunk turns to the left, pointing the way towards worldly success. The various items associated with the deity all have various symbolic meanings.
Images from my trip to Nepal, Autumn 2007.