Christianity developed within Judaism and most of the first converts were Jews with preaching being done in the temple and synagogues. The two were quite similar in that outside observers would have said that Christianity was a sect within Judaism but profound differences did arise over the first two centuries. Judaism tended to be marked by correct practices and ritual whereby Christianity tended toward correct teaching (Orthodoxy and Orthopraxy) Christians tended to reject aspects such as food laws, distinct clothing, strict Sabbath observance and circumcision. Much of the early discussions in the Church were to clarify the relationship between Christianity and Judaism putting the teachings of Jesus first and making him the central figure.
The doctrine that made Christianity distinct from Judaism is the idea that we are liberated through faith and what faith does rather than practicing ritual and concentrating solely on behaviour. This was expressed clearly in terms of the two covenants or testaments where the new testament fulfills what is laid out in the old testament. The old testament writers record Gods dealings with the people of Israel whereas the new testament refers to Gods dealings with humanity as a whole as revealed in the teachings of Jesus. The teachings of Christ usher in a new spiritual life that builds on the laws and requirements of the old testament.
One of the conflicts was around how appropriate the old laws, rituals and customs were and should Gentiles (non Jewish people) observe them. This is highlighted in the book of Acts which details the problems the early Christians encountered in forming a Church and focusing on correct teaching. The book of Acts highlights well the beginnings and spread of the early church including the tensions that arose as Jewish followers, who believed that Jesus was the messiah, insisted that others (Gentiles) carry on with the old rituals particularly circumcision.
The first general Church council in AD 49 considered this issue drawing on the teachings of Jesus and his instruction to not worry about what to eat, what to drink or what to wear. The debate about the need to keep the Judaic law was questioned by Paul who pointed to the rapid growth and growing impact of the new believers and the decision of the council not to require circumcision was a key moment in helping the gentiles come to faith, winning widespread support.
If Christians could only gain salvation through a rigorous observance of the law what was the purpose of Christs death? It was faith in Christ that became the basis of the Christian faith. Paul points out that salvation cannot be attained by good works but rather that works are the fruit that result from having faith in Jesus and that the gift of the holy spirit had not been achieved by the law alone. It is interesting that all the great Jewish patriarchs and prophets had all been justified by faith. The most important outcome from the debate was that Jews and gentiles were to be given equal status and acceptance within the early Church.
Summary and references from Christian history - An Introduction by Alister E. McGrath