The tempters represent
- secular power,
- the desire to fight tyranny, and
- the desire to seek martydom and sainthood for personal, spiritual glory.
Here are the details.
The First Tempter represents the temptation to indulge the senses in parties, entertainment, and light pursuits.
He offers Thomas the prospect of living the social life he had enjoyed in his youth with the king. Laughter, "singing at nightfall" and enjoying "wit, wine, and wisdom." Giving into this wouldn't just be pleasurable, it would be safe. Being "easy" -- offering easy companionship to other pleasure-seekers -- would prevent Thomas from coming into conflict with the king. If Thomas refuses to take this path
"…[Y]our goose may be cooked and eaten to the bone."
The Second Tempter represents the temptation to seek worldly, secular power and the "life lasting" glory that may grow from it:
"...You, master of policy
Whom all acknowledged, should guide the state again."
This temptation isn't simply about obtaining a glorious reputation. The tempter appeals to Thomas's desire to help people in their everyday lives:
"Disarm the ruffian, strengthen the laws
Rule for the good of the better cause…"
This is a more difficult temptation to overcome, but Thomas rejects it. He already possesses the more important power -- spiritual authority. He won't "descend to desire a punier power."
The third tempter is a "rough, straightforward Englishman" representing the barons and the forces that would weaken the king's concentrated power. By succumbing to this tempter, Thomas could join the "fight for liberty" and end the "tyrannous jurisdiction" of the king over the bishops and barons.
Thomas rejects this temptation as well, noting that men who plot against the king are not trustworthy allies, and resolving that "No one shall say that I betrayed the king."
The fourth tempter represents Thomas's desire to become a martyr and saint -- to seek out this fate because it appeals to his pride and selfish wishes for spiritual glory that will outlast his death. This is the most difficult temptation to overcome. Thomas says it's also the "greatest treason / To do the right deed for the wrong reason." The servant of God has the capacity to commit a worse sin than the servant of the king. He can end up using the pretext of doing God's work to further his own personal ends.
The first three tempters offer Becket reward in this life in return for his taking back his judgments against the King. They are, in some ways, reminiscent of the three temptations of Jesus at the start of his ministry, offering him rewards in this world. As with Jesus, Becket rejects these three. The fourth tempter is a bit different because he appeals not to his relationship with the king, but to his own desire to become a saint and a martyr for his work. He is told that without sainthood, he will soon be forgotten. Appealing as this may be, it leads him to the most important decision in the play which he expresses in this way: "The last temptation is the greatest treason:/To do the right deed for the wrong reason." He rejects the temptation to act out of pride and conceit (the wrong reason), and to stay and fight (the right deed.)