In Mark 7:1-13, the Pharisees and scribes accuse Jesus and his disciples of abandoning "the tradition of the elders" because they ate with unwashed hands. This obviously makes Jesus angry, and he turns on them.
"Well did Isaiah prophesy of you hypocrites, as it is written, 'This people honours me with their lips, but their heart is far from me; in vain do they worship me, teaching as doctrines the precepts of men.' You leave the commandment of God and hold fast the tradition of men." And he said to them, "You have a fine way of rejecting the commandment of God, in order to keep your tradition!" For Moses said, 'Honour your father and your mother'; and, 'He who speaks evil of father or mother, let him surely die'; but you say, 'If a man tells his father or his mother, what you would have gained from me is Corban' (that is, given to God) - then you no longer permit him to do anything for his father or mother, thus making void the word of God through your tradition which you hand on."
This story is repeated in Matthew 15:1-9.
How is it then that Tradition is so important for Catholics and Orthodox?
There were some man-made traditions that were concocted by the Pharisees to avoid following God's laws. The best example of this was the one quoted above. There was the "Korban Rule", under which Jews could dedicate all their money to the Temple treasury as an act of thanksgiving. This gave them an excuse of not having any money to help their parents who might come to them in need of help. Callously to ignore your parents' plea for financial help would be to break God's commandment to "honour your father and your mother".
However, the reality was that those who used this "Korban Rule" actually had full access to their money, even though it was officially Temple property. It is easy to see why Christ condemned that tradition as corrupt and contrary to God's justice.
But Our Lord did not condemn tradition as such. The word "tradition" comes from a Latin word meaning "to hand on". Before the story of Jesus was written down, it was a "tradition" - it was "handed on" from one person to another, as each one who heard it retold the story. The whole Bible is part of the "tradition" of the Church because we hand on in written form the history, the stories, the teaching that from generation to generation were handed on by word of mouth. Catholics usually spell this special sort of tradition with a capital 'T', because it is much more important than the sort of tradition, for example, which is the way you celebrate Christmas in your family or where you go for your holidays.
For us God's revelation to his people is preserved in spoken form and written form. The written form we call 'Scripture'. Because this was passed on in spoken form before it was written, Scripture is part of Tradition, something handed on.
We can see this in Luke 1:1-4; "Inasmuch as many have undertaken to compile a narrative of the things which have been accomplished among us, just as they were handed on to us by those who from the beginning were eyewitnesses and ministers of the word, it seemed good to me also, having followed all things closely for some time past, to write an orderly account for you, most excellent Theophilus, that you may know the truth concerning the things of which you have been informed."
Again in John 20:30-31: "Now Jesus did many other signs in the presence of the disciples, which are not written in this book; but these are written that you may believe that Jesus is the Christ, the Son of God, and that believing you may have life in his name."
And again in 1 Thessalonians 2:13: "And we also thank God constantly for this, that when you received the word of God which you heard from us, you accepted it not as the word of men, but as what it really is, the word of God, which is at work in you believers."
Other passages from Scripture:
"But there are also many other things which Jesus did; were every one of them to be written, I suppose that the world itself could not contain the books that would be written." John 21:25
"I commend you because you remember me in everything and maintain the traditions even as I have delivered them to you." 1 Corinthians 11:1
For I received from the Lord what I also handed on to you, that the Lord Jesus on the night when he was betrayed took bread, and when he had given thanks, he broke it, and said, 'This is my body which is for you. Do this in remembrance of me.'" 1 Corinthians 11:23
Remember - these words were written by St. Paul before there was a New Testament. This Tradition came by word of mouth, not from the Bible. Teaching comes to us from God, not just through what is written in the Bible, but also by word of mouth - by what is handed on to us by God's people, the Church!