Short answers, pulled from the story.
Sol Invictus, meaning "Unconquered Sun," was the official sun god of the late Roman Empire. The emperor Aurelian revived and elevated his cult in 274 AD, declaring Sol Invictus the chief god of the empire. His prominence lasted until the emperor Constantine I legalized Christianity and restricted paganism.
Aurelian dedicated a new temple to Sol Invictus on the 25th of December 274, bringing the total number of Sol's temples in Rome to at least four. He also elevated the priesthood of Sol to the senatorial class, replacing lower-ranking sacerdotes with pontifices, and established four-yearly games in the god's honor.
The widely held hypothesis is that the early Church chose December 25 for Christmas to absorb the festival of the Invincible Sun's birthday, known as Dies Natalis Solis Invicti. The Calendar of Filocalus from around 336 AD is the earliest record placing both celebrations on that date. However, scholar Steven Hijmans argues the pagan feast may have been a reaction to the Christian one rather than its source, and an alternative theory proposed by Louis Duchesne in 1889 calculates Christmas as nine months after March 25, the date of Christ's conceived conception.
Constantine used the legend claiming the "Unconquered Sun" as a companion to the emperor with particular frequency. His official coinage bore images of Sol until 325-326 AD, and a gold solidus and gold medallion from his reign show the emperor's profile jugate, or side by side, with Sol Invictus.
This is a longstanding scholarly dispute. The traditional view held that Sol Invictus was a Syrian sun god, possibly Elagabal of Emesa or Malakbel of Palmyra, first introduced to Rome under the emperor Elagabalus. The revisionist view, associated with S. E. Hijmans, holds that there was only one continuous cult of Sol in Rome from the monarchy onward, making Sol Invictus simply an elevated form of the traditional Roman sun god.
The last known dated inscription referring to Sol Invictus is from 387 AD. Despite this, the cult retained enough followers into the fifth century that the Christian theologian Augustine of Hippo found it necessary to preach against its devotees.