The Jewish belief of Jews as a chosen people is that Jews are the chosen people of God. Some Jews believe that God has given them a special job to repair the world and make it a better place. According to this view, they must use the things in the world to increase good and come closer to God, and as God’s partner in repairing the world – to find ways to lessen suffering of people and animals, to make more peace and respect between people, and to protect the earth’s environment from destruction. This process is known as “tikkun olam” (from Hebrew) – repairing the world.

From the Bible
This origin of the idea in the Torah (the first five books of the Tanakh, which are also included in the Christian Bible). Much is written about these topics in rabbinic literature:

According to Judaism, God made an agreement called a “covenant” with Abraham, the ancestor of the Jewish people. The Bible says that God promised to bless Abraham and his descendants if they worshipped God and were faithful to him. God made this covenant with Abraham's son, Isaac, and with Isaac's son, Jacob. God also gave Jacob another name – Israel. This is how Jacob’s descendants got the name the “Children of Israel” or “Israelites.” The Jewish religion, God later gave the Israelites the Torah to the Israelites through their leader, Moses. The Torah told the Israelites how to live and build their community. God gave the Israelites the Ten Commandments and other laws in the Torah.

Interpretations

Religious Jewish views
The Jews are sometimes called the “Chosen People" because the Bible says God told them “you will be to Me a kingdom of priests and a holy nation” (Exodus 19:6) and “For you are a holy people to the Lord your God, and the Lord chose you to be His own special nation out of all peoples on the earth” (Deuteronomy 14:2). Religious Jews ("orthodox") understand this means that they have special duties and responsibilities commanded by God - for example, that they must build a just society and serve only God. They believe that this covenant works in two ways: if they follow God’s laws, He will give them his love and protection, but they are also responsible for their sins – bad actions – and not doing what God told them. Religious Jews believe that they must teach other people that God exists and that God wants all people to do good actions. Jews believe that their job in the world is to be "a light to the nations" (Isiah 49:6) by showing the people of the world ways to make the world a better place.

Modern views
The idea of chosenness has traditionally been interpreted by Jews in two ways: one way is that God chose the Israelites, while the other is that the Israelites chose God. Another opinion is that even though the Jews chose to follow God, the Kabbalah and Tanya teach that even prior to creation, the "Jewish soul" was already chosen. However, Jews do not believe that being a member of the Chosen People gives them any special talents or makes them better than anyone else, and many of whose who are considered "Jewish" (especially secular/(non-religious ones) do not believe they are "Chosen" in any way at all. 

Jews generally do not try to convince other people to believe in Judaism. Jews believe they have a special job to show all peoples that God exists, but people do not have to be Jewish to follow God. All people can serve God by following the Seven Commandments (rules) given to Noah. But, Judaism accepts people who choose to change their religion to Judaism.

From other religions
The Jewish people have a special status in the Islamic book, the Quran:
O children of Israel, remember my favor which I bestowed upon you, and that I favored you above all creation. (Qur'an 2:47). 2:122).

Many Christians also believe that the Jews were God's chosen people (), but because of Jewish rejection of Jesus, the Christians in turn received that special status (). This doctrine is known as Supersessionism.

References

Other websites
The Contemporary Rivalry over the Chosen People: Jewish, Christian, and Muslim Perspectives - Avi Beker at the Institute for Global Jewish Affairs
Chosen people at the Jewish Encyclopedia
The Jewish concept of chosenness
The Chosen People FAQs
How does Jewish Pride differ from Nazi Supremacy?, Rabbi Tzvi Freeman

Judaism