A martyr is someone who is killed for no other reason than for their religion, faith or beliefs.
Persecution is hostility and ill-treatment, especially because of race or political or religious beliefs; oppression.
The ways that early Christians were persecuted were being crucified, stoned to death, decapitated or eaten by lions. They were humiliated in local villages in front of the public in the hands of Roman authorities. Some went out of their way just to persecute Christians.
Early Christians were persecuted because of their faith and beliefs. The guards, priests and the Sadducees didn’t like the idea of Christians ‘making up’ and believing in a new religion, believing that the Messiah is coming. The Romans didn’t want the Christian faith to develop and be historical. They worried that their gods were angry at the Empire's new faithfulness to Christianity. Therefore they didn’t agree with Christianity and chose to persecute them.