Ganesha is the unifying god of all eastern religions. He is accepted by Vaishnavites as well as Shaivites also the Buddhists and Jains, he is an Unifier.

The elephant is the most powerful and intelligent animal in the forest. Neither lions nor tigers attack it. It moves through the forest by smashing its way through the dense growth making a path for other animals. That is why an elephant is symbolic as the remover of obstacles. The elephant symbolizes the qualities of wisdom and faithfulness (a elephant is usually faithful to only one trainer in a lifetime) signifying a quality we need to inculcate as regards parents or spouses. The small eyes symbolize concentration and single mindedness.

It has four or sometimes six arms which symbolizes the controller of the four realms of beings i.e. the sea, land, water (including amphibians), and air. When he has 6 hands it includes the subtle lower and higher realms of existence.

In one hand is a pen or a axe or a chakra which symbolizes knowledge or the power to cut the bonds of ignorance.

In one hand is a OM or a dot (Bindu), which symbolizes we should meditate on the ultimate reality that lies beyond the bindu or point of unity of the universe.

In one is the lotus which grows in stagnant water, but raises its flower above the water line so that its purity remains unblemished by the polluted waters underneath. In just the same way we should live in this world but remain unpolluted by the materialism and negative qualities that permeate our surroundings.

In one of his right hands he holds a hook or Ankusa and in one of his left hands he holds a noose or Paasa. These tools are used by elephant-trainers to tame wild elephants. Symbolically, the mind is like a wild elephant, running from here to there, in chaos. So we must use the ankusa to tame and control our mind and then use the noose to tie it down and keep it close to us - only then can we concentrate, contemplate and meditate.

In the final hand he holds sweets, that symbolize the reward you get when you attain mastery over the elements of other hands .

Lord Ganesha's large belly is said to contain the whole universe - and is symbolic of the way we should live our lives. We must be able to accept and digest whatever experiences we go through - be they good or bad. Throughout our trials and tribulations we must be equanimous and realise that all experiences are learning experiences and will bring us closer to God.

His large ears mean he gains wisdom through listening.

The elephant is vegetarian yet has immense strength, signifying qualities of non-violence despite immense strength. Strength that is always under control and never used for violence. In fact it is the most powerful animal in the forest, demonstrating you don't need to eat meat to become strong.

The elephant's trunk is one of the few instruments in nature that can perform both gross and subtle actions - it can uproot a huge tree as well as pick up the tiniest of needles. It signifies the need for us to be able to develop our intellect, our powers of discrimination, in both the gross or material world and also in the subtle or spiritual world.

Ganesha sits on a rat, the rat is symbolic of a animal that is active in the darkness of the night who sniffs out food by smelling its way around the darkness. Smell in Sanskrit is translated as Vasna which in Sanskrit also means negative mental traits. Thus vasna has a dual meaning, i.e. it symbolizes our negative traits that dominate when ignorance (or darkness) exists.

When Ganesha sits on a rat it means use wisdom and intelligence to trounce and master all the negative mental attributes or vasnas, that take you into the darkness of ignorance and the resultant suffering that comes due to this ignorance.

When Ganesha sits on the rat with one leg on top and one touching the ground, it means focus your mind on the divine existence in the upper realms of the universe and yet have one foot firmly planted on the ground right through life.

There also lies an interesting interpretation in our scriptures based on the life of Ganesha, who is the son of Shiva and Parvati.

Shiva leaves Parvati for a long time on one of his frequent travels. Ganesha grows up with his mother not really knowing who the father is. One day Parvati has a bath in a river and requests Ganesha to make sure no one comes along. As Ganesha stands guard, Shiva comes along searching for Parvati. Ganesha does not recognize his father and prevents Shiva from going to Parvati. Enraged Shiva cuts off Ganeshas head and replaces it with that of an elephant passing by, after Parvati comes rushing hearing the commotion. The symbolism here is, when you are ignorant, you don't recognize even your near and dear ones. Shiva decides to replace it with an elephants head, as it does not have an ego (ahankara) that blinds us to reality. The elephant also symbolizes wisdom in Indian culture.

It is sad most Hindus are very literal today when interpreting Hinduism. The deeper symbolic meanings have been lost over time. Most assume Ganesha is a physical being with an elephant head and a human body running around like a strange creature. Ganesha is symbolic and conveys a important message on how we need to live, and avoid the traps of life that we fall into due to ignorance and the means of overcoming these obstacles.

The Lesson one needs to comprehend is, it is pointless to pray to Ganesha and expect some sort of a physical creature to help you. The key and the manner in which you overcome your obstacles is displayed right in front of your eyes. If you understand those symbols, assimilate and implement them in your life, all the obstacles in your path will disappear. Ganesha is showing you the way, now you have to implement it yourself. Don't expect someone to come and do the physical work of removing your vasnas or obstacles for you.

Also read
1. Ganesha - Global God in a Globalised World
2. Salvation to the King Elephant - Gajendra Moksha - from the Bhagavat - Mahapuran