What are the Ten Commandments?
Key Verse:  These words the LORD spake unto all your assembly in the mount out of the midst of the fire, of the cloud, and of the thick darkness, with a great voice: and he added no more. And he wrote them in two tables of stone, and delivered them unto me.  [Deuteronomy 5:22]
Where did the Ten Commandments come from?
After leading the Children of Israel out of Egypt, Moses brought them to mount Sinai.  While they camped at Sinai, God brought Moses up into the mountain to speak with him.
And he gave unto Moses, when he had made an end of communing with him upon mount Sinai, two tables of testimony, tables of stone, written with the finger of God.  [Exodus 31:18]
And Moses turned, and went down from the mount, and the two tables of the testimony were in his hand: the tables were written on both their sides; on the one side and on the other were they written.  And the tables were the work of God, and the writing was the writing of God, graven upon the tables.  [Exodus 32:15-16]
God gave Moses two tables of stone that God had carved out of the mountain and written his law on.  The story continues . . .
And it came to pass, as soon as he came nigh unto the camp, that he saw the calf, and the dancing: and Moses' anger waxed hot, and he cast the tables out of his hands, and brake them beneath the mount.  [Exodus 32:19]
Moses was so angry at the Children of Israel for breaking God's laws that he smashed the tables on stone on the ground.  God later instructed Moses to carve out two more tables of stone so that the laws could be rewritten.
And the LORD said unto Moses, Hew thee two tables of stone like unto the first: and I will write upon these tables the words that were in the first tables, which thou brakest.  [Exodus 34:1]
God took the two tables of stone that Moses had carved and rewrote on them the same words that were on the first.  The writing on both the first and the second set of stone tablets were the Ten Commandments, spoken by God at mount Sinai. 
These words the LORD spake unto all your assembly in the mount out of the midst of the fire, of the cloud, and of the thick darkness, with a great voice: and he added no more. And he wrote them in two tables of stone, and delivered them unto me.  [Deuteronomy 5:22]
Moses made this statement immediately after reciting the Ten Commandments.  These commandments are recorded both in the 20th chapter of Exodus and the 5th chapter of Deuteronomy.  We will discuss them in greater detail later.
Did Jesus do away with the Ten Commandments?
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 Ten Commandments were given to Moses and the Children of Israel before the  Levitical laws were given and therefore are part of the Melchisedec priesthood.
And it came to pass, as soon as he came nigh unto the camp, that he saw the calf, and the dancing: and Moses' anger waxed hot, and he cast the tables out of his hands, and brake them beneath the mount.  And he took the calf which they had made, and burnt it in the fire, and ground it to powder, and strowed it upon the water, and made the children of Israel drink of it.  [Exodus 32:19-20]
Then Moses stood in the gate of the camp, and said, Who is on the LORD'S side? let him come unto me. And all the sons of Levi gathered themselves together unto him.  [Exodus 32:26]
The Levitical priesthood was established as a result of the transgressions of the Children of Israel at Sinai.  The transgression of worshiping the golden calf, an idol, was a transgression of the second commandment.

Jesus only did away with the Levitical law, which was written by the hand of Moses, not the Ten Commandments which were written by the hand of God.
What laws are contained in the Ten Commandments?
Thou shalt have no other gods before me.  [Exodus 20:3]
Thou shalt have none other gods before me.  [Deuteronomy 5:7]
The first commandment simply states that God is the only god we are to serve.  We are not to worship any other gods.
Thou shalt not make unto thee any graven image, or any likeness of any thing that is in heaven above, or that is in the earth beneath, or that is in the water under the earth:  Thou shalt not bow down thyself to them, nor serve them: for I the LORD thy God am a jealous God, visiting the iniquity of the fathers upon the children unto the third and fourth generation of them that hate me;  And showing mercy unto thousands of them that love me, and keep my commandments.  [Exodus 20:4-6]
Thou shalt not make thee any graven image, or any likeness of any thing that is in heaven above, or that is in the earth beneath, or that is in the waters beneath the earth:  Thou shalt not bow down thyself unto them, nor serve them: for I the LORD thy God am a jealous God, visiting the iniquity of the fathers upon the children unto the third and fourth generation of them that hate me,  And showing mercy unto thousands of them that love me and keep my commandments.  [Deuteronomy 5:8-10]
The second commandment deals with idol worship.  We are not to make idols, bow down to idols, or serve idols.  This is the commandment that the Children of Israel broke when they made and worshiped the golden calf while Moses was on mount Sinai.

This is the only commandment that carries with it a warning within the commandment itself.  God states he will not only punish you, but also your children, your grandchildren, and your great-grandchildren if you break this commandment.  God is very clear that we are to worship only him.
Thou shalt not take the name of the LORD thy God in vain; for the LORD will not hold him guiltless that taketh his name in vain.  [Exodus 20:7]
Thou shalt not take the name of the LORD thy God in vain: for the LORD will not hold him guiltless that taketh his name in vain.  [Deuteronomy 5:11]
The third commandment instructs us in how we are to speak of the Lord.  We are not to take his name in vain, or speak it casually or in swearing.  When we speak God's name, it must be done in a reverant way and it must be respectful to God.  For  example, saying "God" or "Lord" while offering a prayer is a proper way to use God's name.  Shouting it with other explicitives after hitting your thumb with a hammer is breaking this commandment.

Many people don't realize the importance of this commandment.  God put breaking it on the same level as idol worship or murder.  Taking the Lord's name in vain is no different in God's eyes than bowing down to an idol or killing another person.
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]
Keep the sabbath day to sanctify it, as the LORD thy God hath commanded thee.  Six days thou shalt 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, nor thy manservant, nor thy maidservant, nor thine ox, nor thine ass, nor any of thy cattle, nor thy stranger that is within thy gates; that thy manservant and thy maidservant may rest as well as thou.  And remember that thou wast a servant in the land of Egypt, and that the LORD thy God brought thee out thence through a mighty hand and by a stretched out arm: therefore the LORD thy God commanded thee to keep the sabbath day.  [Deuteronomy 5:12-15]
Commandment number four deals with setting the seventh, or Sabbath, day apart from other days of the week.  We are given two instructions on how we are to treat the Sabbath different from other days.  1) We must keep the Sabbath day holy.  2) We are to cease from our labors on the Sabbath day along with everyone residing within our house or under our control.  God made the Sabbath a day for man to rest and worship him.  Remember, God blessed the Sabbath day and hallowed it.

Follow this link for more information on the Sabbath.
Honour thy father and thy mother: that thy days may be long upon the land which the LORD thy God giveth thee.  [Exodus 20:12]
Honour thy father and thy mother, as the LORD thy God hath commanded thee; that thy days may be prolonged, and that it may go well with thee, in the land which the LORD thy God giveth thee.  [Deuteronomy 5:16]
The fifth commandment instructs us to honor, or give respect, to our parents.  This commandment is the only one that comes with a promise.  If we respect our parents, God will grant us a long life here on Earth.
Thou shalt not kill.  [Exodus 20:13]
Thou shalt not kill.  [Deuteronomy 5:17]
This commandment plainly instructs us not to kill our fellow man.
Thou shalt not commit adultery.  [Exodus 20:14]
Neither shalt thou commit adultery.  [Deuteronomy 5:18]
Commandment seven deals with the sanctity of marrage.  We are forbidden from involving ourself with someone who is married and are forbidden from being involved with anyone other than our spouse if we are married.
Thou shalt not steal.  [Exodus 20:15]
Neither shalt thou steal.  [Deuteronomy 5:19]
The eighth commandment tells us that we are not to take what doesn't belong to us.  We are to respect the property of others and are not to take as our own what rightfully belongs to others.
Thou shalt not bear false witness against thy neighbour.  [Exodus 20:16]
Neither shalt thou bear false witness against thy neighbour.  [Deuteronomy 5:20]
Commandment number nine forbids us from making false statements, or telling lies, that harm other people.
Thou shalt not covet thy neighbour's house, thou shalt not covet thy neighbour's wife, nor his manservant, nor his maidservant, nor his ox, nor his ass, nor any thing that is thy neighbour's.  [Exodus 20:17]
Neither shalt thou desire thy neighbour's wife, neither shalt thou covet thy neighbour's house, his field, or his manservant, or his maidservant, his ox, or his ass, or any thing that is thy neighbour's.  [Deuteronomy 5:21]
The tenth and final commandment tells us not to covet, or  desire to the point of wanting to take, another man's possessions.  Not only are we not to take another man's possessions (Commandment #8) or involve ourselves with another's spouse (Commandment #7) but we also are forbidden from desiring to steal or to commit adultery.

For more information on the Ten Commandments, click on the link to The Law of the Lord Bible study tract.
Conclusion
The Ten Commandments were given by God before the establishment of the Levitical priesthood.  They are not part of the law Jesus did away with.  These commandments (along with the law of the clean and unclean meats and tithing) represent God's laws for man and remain in full effect for us today.