Most Christians worship on Sunday today because Constantine made Sunday the Roman day of rest in 321.
"On the venerable day of the Sun let the magistrates and people residing in cities rest, and let all workshops be closed. In the country however persons engaged in agriculture may freely and lawfully continue their pursuits because it often happens that another day is not suitable for grain-sowing or vine planting; lest by neglecting the proper moment for such operations the bounty of heaven should be lost."
He did so, for likely 2 reasons.
1) While many sects kept the sabbath tradition up until the Laodicean Council in 364 (particularly eastern denominations) a good portion of the early church had already given up the Sabbath, just as they had given up many of the laws of the old covenant. Constantine's support of Christianity by making Sunday a secular holiday continued this trend.
St. Ignatius wrote to the Magnesians (~100AD):
We have seen how former adherents of the ancient customs have since attained to a new hope; so that they have given up keeping the Sabbath, and now order their lives by the Lord's Day instead - the Day when life first dawned for us, thanks to Him (Jesus) and His death.
2) It was easier to make Sunday the day of rest, and provided political power to the Roman Church. Due to the existing Sol worship in Rome, and the western Church's existing Lord's Day worship, Sunday was both the easy choice, and it provided the Roman church with power over the eastern, Sabbath-following church. He gave his mother's Christianity a boost over other roman religions, and did so without making anyone (other than the Jews, for the most part) mad about it.
Roman traditions, particularly those which then became standard in the RCC, have influenced western culture heavily ever since.
As for why many of the early Christians gave up the sabbath prior to Emperor Jovian's council and its outlawing of sabbath keeping, they also gave up circumcision, washing of hands before eating, stoning people for working on Saturday, etc, etc, etc. The new covenant between man and god through Jesus removed the need to follow the law of the sabbath, so long as the holiness of the sabbath was kept.
The selection of Sunday (instead of, say Wednesday), seems to stem from Jesus' resurrection occurring on "the first day of the week" (John 20:1) and possibly more importantly, the Pentecost/birth of the church 50 days later (Acts 2:1) - also a Sunday. Further reasoning as to why Sunday was chosen is potentially up for debate, but I'm not aware of much other than circumstantial evidence to suggest that it was specifically sun worship. The only thing hinting at such a thing, in my mind, is the increase prominence of the Lord's Day in the Roman & Greek areas, whereas the eastern regions tended to hold to the Saturday tradition; and that seems merely a regional variation, possibly because it was easier for a non-Jewish Roman to convert from one Sunday-based religion to another: no need to alter your work schedule.
