The bells signify the silver trumpets by which people of the Old Law were called to the sacrifice. They also signify the priest preaching. The Lord commanded Moses to make a vestment for the high priest and ordered seventy-two bells to sound when the high priest entered into the Holy of Holies (Ex. 28:35). This sounding of the bell exhorts the faithful to pay attention to the preacher. But the cavity of the bell reminds the preacher of humility and of the warning of St. Paul: "I am become as sounding brass or a tinkling symbol" (1 Cor. 13:1).
The hardness of the metal signifies the fortitude that should be in the mind of the preacher; "Behold I have made thy face strong against their faces" (Ez. 3:8).
The clapper that strikes the two sides of the bell is the tongue of the preacher, who should preach from both the Old and New Testaments. A bishop who cannot preach is like a bell without a clapper. St. Gregory the Great calls him a dog that cannot bark!
The striking of the bell signifies the striking of vices in the preacher, and because the word of God is a two-edged sword, the sermon is first for the correction of the preacher ("lest having preached to others, he himself should be a castaway" [cf. 1 Cor. 9:27]) and, second, for the correction of the faithful.
The link by which the clapper is joined is moderation, by which the tongue of the preacher should be ruled.
The wood of the frame on which the bell is fastened stands for the wood of the Cross, lifted high. The rope to pull the bell comes from this wood, which reminds the preacher that he must preach Christ and Him crucified (cf. 1 Cor. 1:23). A rope is usually composed of three strands, so preaching should comprise morality, allegory, and history. The rope goes high and low when it is rung, for sometimes the preacher should speak of lofty things, and at other times of simple things, as St. Paul said: "Whether we exalt ourselves it is for God, or whether we humble ourselves it is for you." (2 Cor. 5:13) When the priest pulls the rope downward, he descends from contemplation to the active life. But when, holding on to the rope, he is drawn upward, he is raised in contemplation.
In a Catholic culture, the bells were usually rung twelve times a day, since the day is canonically divided into two parts of twelve hours each. These correspond to the Divine Office. This scheme of ringing the bells was meant to sound the death knell of the Passion and death of Christ.
At Matins and Lauds (6am) He is arrested and bound. At Prime (8am) He is reviled. At Terce (9am, the third canonical hour) He is condemned to death. At Sext (Noon, the sixth canonical hour) He is nailed to the Cross. At None (3pm, the ninth canonical hour; notice the connection here with the Hour of Divine Mercy) His side is pierced. At Vespers (6pm) He is taken down from the Cross, and at Compline (9pm) He is laid to rest. The bells can also be rung three times a day at the hours of the Angelus at 6am, Noon and 6pm in honour of the Incarnation and of the Holy Trinity.
There was quite a science to the ringing of bells. In English the words ringing, tolling, knelling, chiming, and chanting all refer to different ways of sounding the bells. This art has largely been lost, and I pray that someday it will return.
Bells were also rung when someone was dying, so that the parish would know to pray for the person. This was called the Passing Bell. Bells were rung for processions so that evil spirits might hear and flee, as a tyrant hearing the trumpets of the king coming to depose him would flee in fear.
Bells were silent in case of an interdict (an ecclesiastical punishment for a congregation or a town): "I will make thy tongue cleave to the roof of thy mouth, for they are a rebellious house" (Ez. 3:26). This was considered a severe punishment.