The ring on his right hand has various special powers; it functioned something like a sonic screwdriver, only before the sonic screwdriver entered the show. From Wikia:
The Doctor wore the ring on the middle finger of his right hand in his first incarnation, but soon discarded it after his regeneration into his second incarnation, as it did not fit him any longer.
The ring had several special abilities. It could supply power to the TARDIS's main doors and control a Zarbi.  If given the correct spectrum of light, it could unjam a TARDIS's locking mechanism.  The ring seemed to be able to facilitate hypnotism and protect the Doctor from electrical shocks. 
And in non-TV canon:
On the Ethereal homeworld, the Doctor used the ring to open an electronic lock.  When the Doctor received lessons in the 'magic' of Avalon from Kilvenny Odoyle, the ring served as his 'wand' through which he focused his spells. 
The ring on his left hand, however, doesn't seem to have any significance.
As far as I can tell, it was never mentioned on the show or any special attention drawn to it.
As an interesting side note, the Twelfth and current Doctor is wearing a new ring on the third finger of his left hand, which may also have some significance, given that it was specially commissioned for him and came with the following note:
This timeless gold ring contains green amber collected by the Doctor on the planet Raxacoricofallapatorius. It reminds him of all the wars he has won but also all the lives he has taken. This thought lurks on his conscience to this day. To Peter, Best wishes being the New Doctor Who from Idris