Judaism is the religion out of which Christianity was born (Jesus was himself Jewish) and the two share many common features. There are also many differences, some of which I will be covering briefly, the most significant of which is the answer to the question of who is Jesus and was he the messiah promised in the Bible.
Judaism is one of the world's oldest religions, dating back up to 2000 years BC from the time when God first called Abraham to leave his home and follow Him.
At that time, God made a covenant (or agreement) with Abraham in which He promised to make Abraham the father of a great nation and that one day his descendants would inherit the land of Canaan if Abraham followed him.
God's plan was gradually revealed through the Old Testament and built on with further promises to Moses, David and the prophets. This plan in essence is to undo many of the effects of the fall (the sin of Adam and Eve in Eden which led to the loss of God's initial blessing and made man God's enemy). The key elements of this were:
Through Moses, God gave the law to the people of Israel and the fulfilment of the promises became conditional on obedience to the law. To Jews, the law is called the Torah and comprises the written law (basically the Old Testament) and the oral law which Jews believe is handed down from Moses. There are 613 commandments to obey in the Torah of which the 10 Commandments are the best known.
However, Israel was never really able to keep her side of the covenant and be obedient to God' law. Although the people of Israel took possession of the land of Canaan (today's Israel) they continued to rebel against God and finally the people of Israel were exiled to Babylon. During this time, God sent many prophets who warned the people of their sin and reminded them of God's promises. They also added that the exile would not last forever as God would rescue His people and that afterwards the people would be able to obey God and so not be judged again. This is where the promise comes in of a messiah (or anointed one) who would be the rescuer sent from God to fulfil the promises.
The Old Testament ends with Israel waiting for the messiah to come.
Up to this point, Christians and Jews are pretty much in agreement.
Indeed, Christians and Jews share lots of common heritage – after all Jesus was himself Jewish. This leads to many things in common between Judaism and Christianity which include:
However, there are many things different between Judaism and Christianity, of which by far the most significant is the answer to the question of who is Jesus and consequently understanding how God is working out His promises. Jews do not accept Jesus as the promised messiah. Divisions about him started during Jesus' lifetime and continue today.
Jews believe that the messiah will be a human, non-divine person who will restore the physical kingdom of Israel, rebuild the temple in Jerusalem and bring earthly peace. This is very different from the Christian belief that the people of Israel now means all those who follow Jesus and who by faith are Abraham's descendants. Christians believe that Jesus has fulfilled God's promises and has already ushered in the way to a new relationship with God by grace. We acknowledge our own inability to live by God's standards, we trust in Jesus to have paid the price our sin deserves and we find a new path to eternal life through him.
The other key difference between the two faiths is about sin and atonement meaning man's disobedience to God and the means of achieving forgiveness. In the past, in order to receive forgiveness from their sins, the people of Israel had to offer sincere prayers of repentance, turn away from their sin and offer appropriate sacrifices in the temple in Jerusalem. Since the temple was destroyed in AD70 by the Romans, the sacrificial part of this is no longer possible. Jews today believe that the sincere prayers of repentance and turning away from sin are sufficient and would quote 2 Chronicles 7:14 as support of this:
“If my people, who are called by my name humble themselves, and pray and seek my face and turn from their wicked ways, then I will hear from heaven and will forgive their sin and heal their land”.
Our own experience however is that however hard we try, we cannot on our own turn from our wicked ways. Christians believe that Jesus was the only person ever to fully obey God's commands and that he is therefore the only person who didn't deserve God's righteous judgement and punishment of death. His death for us is the only sacrifice sufficient to pay the price for all our sins. Indeed, Christians see the escape of the people of Israel from slavery in Egypt as the biblical illustration of how God planned to rescue people from slavery to sin through Jesus.
There are many other differences in understanding and approach, which there isn't time to cover now.
So, to summarise, Judaism and Christianity share many facets, including a common heritage and view of the Old Testament. There is a fundamental division over Jesus' identity and the implication for how God is working out his plan for mankind.