The developed, private, repeatable penance familiar from later Catholic and Orthodox practice was not how the early church handled serious sin, and the difference is stark enough to be worth describing on its own.
For roughly the first four centuries, many churches practiced exomologesis, an order of public penance available, in the strictest sense, only once in a lifetime, for grave sins, chiefly apostasy, murder and adultery. A penitent confessed to the bishop, was enrolled in an order of penitents, and underwent a period, sometimes years, of exclusion from communion, distinctive dress, fasting and public humiliation, standing at the church door or among the penitents at the back, before formal reconciliation and readmission to the eucharist, often at Easter.
The system produced its own crisis. Because it could be used only once, some Christians delayed baptism itself until near death to avoid the risk of needing penance they could not receive again, a practice Constantine himself is reported to have followed. The rigor of the system was also directly implicated in the Novatianist and later Donatist schisms, both arising from disputes over whether and how those who had lapsed under persecution could be readmitted at all.
The transition to private, repeatable confession is credited to Irish and British monastic practice, using penitential handbooks that assigned specific fixed penances to specific sins, itself a controversial innovation when it spread to the continent from the sixth century, precisely because it broke with a public, once-only model that had stood for centuries and looked to some contemporaries like the church going soft on sin rather than developing its pastoral care.