Ganesha's trunk

Elephants trunks are strong enough to pick up huge logs, but are delicate enough to grasp a single strand of grass.

Ganesha can use his trunk to carry out small, delicate tasks as well as big ones - it is a symbol to show that he can adapt intelligently to all sorts of different tasks and situations.

Ganesha's mouth

Ganesha is sometimes shown with a very small mouth which shows that he talks very little. This means he can listen to the world more carefully.

Other images of Ganesha show a wide mouth. In this case the size of the mouth shows that Ganesha can take in everything that the world as to offer and enjoy it.

Ganesha's eyes

Elephants eyes seem very small in relation to their great size.

Ganesha's small eyes show that he can concentrate very hard.

Ganesha' tusks

Ganesha seems to have one whole tusk and one broken tusk. 

There are many different stories to explain how one of his tusks became broken.

Here is one:

One day, one of Shiva followers came to visit him but, since Shiva was sleeping, Ganesha would not let him in. In the end there was a fight and the man fell on the ground. Eventually he picked himself up and threw an axe at Ganesha. When Ganesha recognised that the axe flying towards him was one that Shiva had given to the man, Ganesha let the axe hit him on one of his tusks which immediately broke.

Here is another story:

Ganesha was asked to write down an important story. Realising how important his task was and that an ordinary pen would not do, Ganesha broke off one of his tusks and made a pen out of it. 

The moral of this story is said to be that "no sacrifice is big enough when you are seeking knowledge and wisdom".

Ganesha: man, elephant, serpant and mouse

Images of Ganesha almost always include parts of four animals - man, elephant, serpant (or snake) and a mouse (or rat).

Ganesha is supposed to ride on the mouse.

This may seem strange, but mice are supposed to be clever animals - just like elephants - but are wondering, inquisitive animals. Showing a mouse as servant of Ganesha shows that our minds can be tamed by the power of Ganesha's love and understanding.

Ganesha has a serpant or snake as a pet. Many people are afraid of snakes, but Ganesha tells us that we can rise above our fears and problems.

The mouse

The little mouse whom Ganesha is supposed to ride upon is another enigmatic feature in his iconography. At
a first glance it seems strange that the lord of wisdom has been granted a humble obsequious mouse quite incapable of lifting the bulging belly and massive head that he possesses. But it implies that wisdom is an attribute of ugly conglomeration of factors and further that the wise do not find anything in the world  disproportionate or ugly.

The mouse is, in every respect, comparable to the intellect. It is able to slip unobserved or without our knowledge into places which we would have not thought it possible to penetrate. In doing this it is hardly concerned whether it is seeking virtue or vice. The mouse thus represents our wandering, wayward mind,
lured to undesirable or corrupting grounds. By showing the mouse paying subservience to Lord Ganesha it is implied that the intellect has been tamed through Ganesha's power of discrimination.

AUM

No analysis of Lord Ganesha can be concluded without a mention of the mystical syllable AUM. The sacred AUM is the most powerful Universal symbol of the divine presence in Hindu thought. It is further said to be the sound which was generated when the world first came into being. The written manifestation of this divine symbol when inverted gives the perfect profile of the god with the elephant head.

Ganesha is thus the ONLY god to be associated in a "physical" sense with the primordial sacred sound AUM, a telling reminder of his supreme position in the Hindu pantheon.