The ancient rishis were so deeply intelligent that they chose to express Divinity in terms of symbols rather than words, since words change over time, but symbols remain unchanged. When we worship Lord Ganesha the qualities that he represents get kindled within us.
Gana : Signifies the ultimate truth that this existing world is nothing but a collection of molecules. This is called as ‘Gana’ (collective form). Our own body is a ‘Gana’. It is made up of flesh, blood and bone marrow. Thus the Lord of all 'Ganas' is Ganesha.
Elephant-head: Signifies authority, endurance, strength and courage.
Mouse as a vehicle: Signifies the mouse nibbling away at ropes that bind. Just like a mantra which can cut through sheets and sheets of ignorance and carry even an elephant through.
Big Belly: Signifies generosity and total acceptance.
Single tusk: Signifies one-pointedness.
Upraised hand : Depicts protection.
Lowered hand : Signifies endless giving and and also symbolizes the fact that we will all dissolve into the earth one day.
Riddhi & Siddhi wives of Lord Ganesha: Signifies that both Riddhi (intelligence) and Siddhi(enhanced abilities) go together with wisdom. Lord Ganesha is considered to be the Lord of Wisdom.
Ankusha and noose : The Ankusha (the goad/stick that is used to prod an elephant awake) signifies ‘awakening’ and the ‘Paasa’ (the noose) which signifies control. Together they signify that with inner-awakening, a lot of energy is released which can go haywire without proper guidance (control) .
Modak: The ‘Modak’ in Ganesha’s hand is the attainment of the ‘Ultimate Bliss’.