The Sabbath was instituted to commemorate the creation of the world, by honoring the seventh “day” or creationary period of time, whereon the Lord rested following His labors over the six preceding “days.”
In our method of calendaring, this made the day we call Saturday the Sabbath.
Later, the Sabbath was changed slightly by a commandment also to memorialize the exodus from Egypt, though Saturday remained the appointed day for reverent worship and rest. (Deuteronomy 5: 15)
This schedule continued through many centuries leading up to the "Meridian of time," marked by Jesus's mortal life upon the earth.
In the early days of Jesus’s dispensation, Christians worshipped on the “Lord’s day,” to remember His resurrection from the dead.
This was on Sundays, corresponding to the day following the Sabbath Saturday He spent resting in the spirit world, after being crucified and dying on Friday. This Sunday was the “third day” as the people measured time, by counting Friday, Saturday, and Sunday.
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Though the foregoing Biblical context is helpful, we don’t rely on any previous dispensation for complete instruction regarding our Sabbath.
We have been given clear direction in our own time, by a specific revelation to the Prophet, Joseph Smith. The revelation was given on August 7, 1831, a Sunday.
Here is the revelation, in part:
"And that thou mayest more fully keep thyself unspotted from the world, thou shalt go to the house of prayer and offer up thy sacraments upon my holy day;
For verily this is a day appointed unto you to rest from your labors, and to pay thy devotions unto the Most High;
Nevertheless thy vows shall be offered up in righteousness on all days and at all times;
But remember that on this, the Lord’s day, thou shalt offer thine oblations and thy sacraments unto the Most High, confessing thy sins unto thy brethren, and before the Lord.
And on this day thou shalt do none other thing, only let thy food be prepared with singleness of heart that thy fasting may be perfect, or, in other words, that thy joy may be full."
Doc. & Cov. 59: 9-13, italics added
If we understand the phrase “on this, the Lord’s day” to be a reference to the actual day the revelation was given (August 7, 1831), it clearly indicates that Sunday is the appointed day of the week to be honored as the Sabbath in our current dispensation.
