Short answers, pulled from the story.
The Decian persecution began on the 3rd of January 250 when Emperor Decius issued his decree. The enforcement of the edict died down completely by 251, lasting approximately eighteen months before Decius himself died in June 251.
A written certificate called a libellus served as physical proof of loyalty required by the empire under Decius. Every person except Jews had to sacrifice and burn incense to the gods in the presence of a Roman magistrate who would sign the document.
Roman authorities granted Judaism legal standing as a religio licita or permitted religion following policies formulated by Julius Caesar and extended by Augustus. This exemption did not apply to Christians because they were viewed as a new phenomenon rather than a recognized religion with ancestral traditions.
Refusal to obtain a certificate resulted in the deaths of Pope Fabian, Babylas of Antioch, and Alexander of Jerusalem. Many other Christians including Dionisius of Alexandria and Cyprianus of Carthage went into hiding across Egypt, Africa, and Anatolia instead of complying.
The edict remained in force for approximately eighteen months starting from January 250 until Decius died in June 251. By 251 efforts to enforce the edict had died down completely causing his decree to lapse after its short duration.