First things first: we get the lowdown on Thomas More, who, in case you missed that day in history class, is a real dude. In fact, this whole story is, as they say, based on a true story. So if you're feeling a need to brush up on your 16th-century European history, you're in luck.
This royal adviser is already refusing to help Cardinal Wolsey pressure the Catholic Church into granting King Henry VIII a divorce. When Wolsey dies, More becomes Henry's new Chancellor—even though he's even less willing to help the king secure a divorce than Wolsey was.
The king visits More and isn't psyched when More demonstrates his unwillingness to break with the Church. Meanwhile, a sniveling upstart named Richard Rich (not to be confused with Richie Rich) is considering trying to help the shrewd adviser, Thomas Cromwell, in bringing More down.
When Henry decides to leave the Catholic Church in order to get his divorce, remarry, and form his own church, More won't go with him. He resigns his post as Chancellor, which passes to his devious rival, Cromwell. When More refuses to sign an oath swearing that the king is now the head of the Church—while otherwise remaining silent—Cromwell goes after him.
More is imprisoned for high treason and interrogated by Cromwell and More's old friend, the Duke of Norfolk. But neither Norfolk nor More's own loving family members can convince him to sign the oath.
Dude is stubborn.
Finally, More is charged with treason in court, where he has to defend himself against Cromwell's arguments. He seems to be doing pretty well in his own defense.
Richard Rich lies, claiming that More openly contested the king's supremacy over the Church in conversation. More's fate is sealed: he'll be put to death. Now free to speak, he eloquently denounces the act of Parliament that authorized the split with the Catholic Church.
Finally, More is executed by beheading, and he submits to his death peacefully and bravely, confident God will accept him. A voiceover at the end explains that More's rival Cromwell was later executed for treason, too—although his greatest betrayer, Richard Rich, died in bed of natural causes.
So that's anticlimactic.