The painting depicts three male heads at right angles to each other, each sat above an animal. From left to right you will see a wolf, lion and a dog. The men above them purely represent the three ages of man, with a young, a middle aged and an elderly man looking away from each other. Three Ages of Woman by Gustav Klimt is a related work which appeared many centuries later.
There is an inscription at the top of the painting in Latin which translates as "...From the experience of the past, the present acts prudently, lest it spoil future actions...". This clearly relates to the three males figures that sit in the middle of the painting. There are some who believe that Titian is reflecting on his own life within this work, explaining that mistakes earlier in his life were damaging his time as an elderly man.
Another theory is that it depicts an elderly Titian, the artist's son Orazio, plus cousin, Marco Vecellio as the young man. It is a reminder of the importance of handling Titian's will in a fair and respectful manner. There is little chance of a clear decision between these or any other theories being made today due to the years that have passed since Titian's career took hold.
The National Gallery, which owns this painting, describes the triple-headed beast at the foot of the painting with a wolf, a lion and a dog as being a symbol of prudence. Titian received help from his studio in completing this painting but he would have decided its entire composition and also put the key elements in himself. The three headed creature is another nod to mythological themes, referring to the Egyptian god Serapis in this case. Writers from as early as the 5th century, including Macrobius, have explained the meaning of each of the three elements to this beastly creature.