There is no commandment in the New Testament about when we should meet, only that we should meet regularly (Hebrews 10:25). So Christians have chosen to meet at different times and on different days depending on what is convenient. For example, Christians living in Muslim countries often meet on Fridays because that is the Muslim day of rest. But most Christians throughout history have met on Sundays. This is partly out of convenience (most of us don’t have to work on Sundays) and partly based on the historical tradition.
The practice of Christian worship on a Sunday goes back to Bible times. Initially, all Christians were Jews and they went to the synagogue on Saturday (the Sabbath). Perhaps to avoid the Sabbath, they held their Christian services on Sundays (or possibly after sunset on Saturday).
There are a couple of hints of this practice in the New Testament. When in Troas, the church met to break bread (i.e., to share the bread and wine as Jesus commanded) on Sunday:
On the first day of the week, when we were gathered together to break bread, Paul talked with them, intending to depart on the next day, and he prolonged his speech until midnight. (Acts 20:7)
Another possible hint is in 1 Corinthians where Paul instructs the believers in Corinth:
On the first day of every week, each of you is to put something aside and store it up, as he may prosper, so that there will be no collecting when I come. (1 Cor 16:2)
Probably this meant that they were to take up a collection at their regular Sunday meetings.
Outside the New Testament, there are also several references to this practice. The Didache is an early document from approximately 100 AD which describes some of the activities of the Christian churches of that time.
Didache 14: But on the Lord’s Day, after that you have assembled together, break bread and give thanks, having in addition confessed your sins, that your sacrifice may be pure.
The “Lord’s Day” refers to Sunday, probably because it was the day of Jesus’ resurrection (John 20:1). Some clear early references to the “Lord’s Day” meaning Sunday are in the apocryphal Gospel of Peter (c.150 AD), the letter of Dionysius, Bishop of Corinth (c.170AD), the apocryphal Acts of Peter (c.190AD), the Acts of Paul (c.190AD), the Didascalia (3rd century) and in Apostolic Constitutions (4th century).