Just as the Crown is a symbol of the State and its authority, the papal ring symbolizes the church and its authority. The first use of the ring appears to be with Pope Clement IV in 1265.
The Vatican provides this description of the meaning of the ring:
It was customary since the early days of the catholic Church that the faithful would kneel and kiss the ring of the Church's representatives. This courtesy to the Pope reflects the beliefs that the Pope who is the successor of Peter who was tasked by Jesus to establish his Church on earth
The priests are the vessels through which Jesus Christ manifests himself to lead His Church. By kneeling before these chosen Representatives and kissing their rings, (the symbol of authority), the believer acknowledges that they were appointed by Jesus Himself and shows his adoration of Christ. This is a symbolic gesture of respect to an authority figure and it is no different really than when the British stand up when the Queen of England enters a room.
On the other hand, the parallelism of headgear and its symbol of authority may be diminished by the fact that the papal "crown" has changed over time. It was once referred to as a tiara and the the triregnum, but now, with the Pope's wishes, it is a simple mitre. This evolution to the mitre came about as recently incoming popes determined that the coronation was not how their papacies should begin.
This abandonment of the tiara has come with some controversy, as many traditionalist Catholics still consider the tiara "to be one of the most striking symbols of the papacy." (Wikipedia)