Ganeshji is the remover of obstacles... Obstacles are there because there is no Knowledge and awareness... as soon as wisdom dawns, all obstacles and problems just vanish like dew in the sunlight...And really what obstacle can stand in the way of an elephant?!
He has a huge stomach which signifies an all accepting nature... He is the Lord of all kinds of groups (Gan means group), so He accepts everyone... but He also wears a snake as a belt around the stomach, which means that He accepts you, but with the full awareness of a snake! It is not passive chalta hai attitude acceptance, but an active, participative acceptance...
He has many hands, one holds a rope, another a poke and still another a modak (yummy sweet).
He offers you the modak (sweet) of Wisdom and Knowledge and when you come near to Him to take it, He quickly ties you with the rope (probably while you are eating it)... meaning that once Wisdom has dawned, you can never really go back to whatever you used to be!
And it may happen that even having all the Knowledge and Wisdom, you may feel low, unhappy or miserable... then He pokes you (remember the snake? He accepts you as you are, doesn't cast you away, but takes action after accpetance)... to wake you up to Reality... And sometimes if the poking is too much for you and you try to run away (you can't He has got you tied with the rope! :)), He gives you the sweet, the Knowledge :) and that brings you back to Him. And so Lord Ganesha binds you sweetly and firmly to Himself!
He is Vinayaka. The peerless leader. He reports to no one!
And the mouse?
The mouse nibbles the veils of ignorance that surround our consiousness, so that true Knowledge can get established in the Self. So, Lord Ganesha rides the mouse who eats away the anger, the ignorance and allows the Bliss of Knowledge to dawn...
Thus Lord Ganesha is the gift of Lord Shiva for Parvati to teach and nurture her to always be in her true nature... The Celebration and Bliss of the Self!