There are many separate views to the matter. I'll try to list the most common ones, along with sources (at the end of each section). I'll first just describe the views briefly; the sources are available for those more interested.

Catholic

The Catholic Church teaches that the Law of Moses (the Old Law) is a preparation for the Gospel, and as such no longer binding. The New Law (the Law of Gospel) is a perfection of it, given through faith in Christ.

The Law of the Gospel is presented in the Sermon of the Mount (Matthew 5-7), but also in the moral catechesis of the apostolic teachings, such as Romans 12-15, 1 Corinthians 12-13, Colossians 3-4, Ephesians 4-5, etc.

God's Law has always remained the same, and has always been available as the natural moral law (e.g. through conscience), but only Jesus could express it perfectly.

Source: Catechism of the Catholic Church, Second edition. Paragraphs 577-582 and 1950-1986. http://www.scborromeo.org/ccc/ccc_toc2.htm

Reformed

The reformed teaching is that the law was given in three parts:

• The moral law, which God gave first to Adam, and later to Moses on Sinai (the Ten Commandments).
• Ceremonial laws, given to the people of Israel, prefiguring Christ.
• Judicial laws, given to the State of Israel.

The moral law is eternal, binding Adam, the first man, as well as any Christian today. Ceremonial laws were abolished in the New Testament. Judicial laws only concerned the State of Israel.

As noted, the moral law was delivered to Moses as the Ten Commandments. Thus the Ten Commandments apply to Christians today as such. Of other laws in the OT, the moral component applies.

Source: Westminster Confession, chapter XIX. http://www.reformed.org/documents/wcf_with_proofs/ch_XIX.html

Theonomy

Theonomy literally means "God's law". It is the notion that God's law is eternal and universally binding. That is, the entire Bible applies to Christians today.

This doesn't mean that salvation comes by keeping the law, as theonomy is often misrepresented. Salvation is granted "solely by grace through faith in the person and work of Jesus Christ".

Psalm 119:97-98 (KJV) is a good verse to show what theonomy is about:

O how love I thy law! it is my meditation all the day.
Thou through thy commandments hast made me wiser than mine enemies: for they are ever with me.

Source: Duncan, T.M. Theonomy: What it is; what it is not. http://www.ipc.faithweb.com/documents/THEONOMY.htm

Dispensationalism

Dispensationalism makes a distinction between Israel and the Christian church in God's plan. Its major objective is to be able to interpret the Bible consistently literally.

Dispensationalists recognize seven dispensations in the Bible:

1. Innocence (Genesis 1:1–3:7)
2. Conscience (Genesis 3:8–8:22)
3. Human government (Genesis 9:1–11:32)
4. Promise (Genesis 12:1–Exodus 19:25)
5. Law (Exodus 20:1–Acts 2:4)
6. Grace (Acts 2:4–Revelation 20:3)
7. The millennial kingdom (Revelation 20:4-6)

The different dispensations are different ways that God has related to people. Salvation has always been through faith.

As today's Christians are living during the dispensation of grace, or during the New Covenant, the old laws or the Old Covenant do not bind them. Because God and His will do not change, the moral law of the New Covenant hasn't notably changed from the Old Covenant.

Source: What is dispensationalism and is it biblical? http://www.gotquestions.org/dispensationalism.html ^{(I'd like to find a better/more original source)}