What day of the week are we to worship the Lord?
Key Verse:  For in six days the LORD made heaven and earth, the sea, and all that in them is, and rested the seventh day: wherefore the LORD blessed the sabbath day, and hallowed it.  [Exodus 20:11]

Did God set one day apart from the others?
The second chapter of Genesis begins:
Thus the heavens and the earth were finished, and all the host of them.  And on the seventh day God ended his work which he had made; and he rested on the seventh day from all his work which he had made.  And God blessed the seventh day, and sanctified it: because that in it he had rested from all his work which God created and made.  [Genesis 2:1-3]
It is clear that from the beginning of creation that God set the Sabbath day apart to be a day of rest.  This is also stated in the fourth commandment.
Remember the sabbath day, to keep it holy.  Six days shalt thou labour, and do all thy work:  But the seventh day is the sabbath of the LORD thy God: in it thou shalt not do any work, thou, nor thy son, nor thy daughter, thy manservant, nor thy maidservant, nor thy cattle, nor thy stranger that is within thy gates:  For in six days the LORD made heaven and earth, the sea, and all that in them is, and rested the seventh day: wherefore the LORD blessed the sabbath day, and hallowed it.  [Exodus 20:8-11]
These verses make three things clear about the Sabbath.  1) We are to keep the Sabbath holy.  2) The seventh day is the Lord's Sabbath, the day he blessed.  3) Neither we, nor anyone in our house, are to work on the Sabbath day.  The Sabbath day was created for man as a day of rest and worship.
And he said unto them, The sabbath was made for man, and not man for the sabbath:  [Mark 2:27]
God realized that man needed a weekly day of rest when he created him.  He therefore set the Sabbath apart to be that day of rest and also to be a holy day in which we can worship and strengthen our faith. 
What day of the week does the Sabbath fall on today?
To answer this question let us first read of Jesus's actions in the gospel of Luke:
And he came to Nazareth, where he had been brought up: and, as his custom was, he went into the synagogue on the sabbath day, and stood up for to read.  [Luke 4:16]
The key phrase in this verse is "as his custom was".  It is clear that Jesus attended church (synagogue) every Sabbath day.  The day that Jesus kept and that God set aside was kept by the children of Israel in the days of Moses, in the days when Jesus walked the Earth, and is still kept today by members of the Jewish faith.  That day of the week is Saturday and has been carefully followed for the last 3000+ years by the Jews.
Was the Sabbath done away with when Jesus died on the cross?
To answer this question we need to study the Priesthood under which we live.  When Jesus died for our sins on the cross, he did away with the "law contained in ordinances" or the "book of the law" and was established as a high priest after the order of Melchisedec.  The law that Jesus did away with was the added law that was given to the children of Israel under the Levitical priesthood because of transgression.  However the laws that were established under the Melchisedec priesthood before the Levitical laws were given still remain in effect today.  The Sabbath day was given before the Levitical priesthood immediately after creation and still remains in effect as part of the Melchisedec priesthood today.  Paul's letter to the Hebrews makes this clear.
For we which have believed do enter into rest, as he said, As I have sworn in my wrath, if they shall enter into my rest: although the works were finished from the foundation of the world.  For he spake in a certain place of the seventh day on this wise, And God did rest the seventh day from all his works.  And in this place again, If they shall enter into my rest.  Seeing therefore it remaineth that some must enter therein, and they to whom it was first preached entered not in because of unbelief:  Again, he limiteth a certain day, saying in David, To day, after so long a time; as it is said, To day if ye will hear his voice, harden not your hearts.  For if Jesus had given them rest, then would he not afterward have spoken of another day.  There remaineth therefore a rest to the people of God.  For he that is entered into his rest, he also hath ceased from his own works, as God did from his.  Let us labour therefore to enter into that rest, lest any man fall after the same example of unbelief.  [Hebrews 4:3-11]
Why then do most churches worship on Sunday?
There is no biblical evidence that the day of rest that God established was changed from Saturday to Sunday.  This is because this change occured well after the writing of the books in the New Testiment.  The Roman Catholic Church was established in 325AD at the council of Nicaea and they didn't want any part of Judaism in their church.  They forbaid Christians from "Judaizing by resting on the Sabbath".

The effect of this council, backed by the force of Roman law, cause a majority of churchs at that time to switch from Saturday to Sunday as their day of rest and worship.  We follow the teaching of the Bible as the unique, inspired word of God and the Bible teaches that Saturday is the Sabbath.

See the tracts The Change of the Sabbath and The Lord's Santified Day for a more in-depth discussion on this subject.
The Sabbath day, which falls on Saturday, was a covenant established by God for man from the creation.  It is a day created for man to rest and worship God.  God's commandment concerning the Sabbath was established at the creation and is still in effect for us today.