A testament is the declaration of a will that has to be executed unalterably upon the death of the testator. When God sent his son Jesus into this world, he expressed his will that all humans who believe in the son shall be saved into eternal life. By the death of the Lord Jesus this will got a binding character that cannot be changed, just like a testament becomes valid by the death of the testator. Because the documents in question here are related primarily to God’s saviour attitude and the death of the Lord Jesus, they can be called righteously a testament. (One should not be troubled by the fact that nowadays a testament may have a different “look and feel”.)
New is the testament of the Lord Jesus insofar that it forms a clear and assertive contrast to those teachings that had been valid before. Those former teachings were based on the Divine Law that had been proclaimed to the people of Israel through Moses the Prophet. Although Jesus confirmed the validity of the Mosaic Law for the Jews, he corrected its (mis)interpretation that had been practiced then. Furthermore, his teaching pointed much more clearly towards God’s Grace as grounds for any justification than the Old Testament had done. Therefore we find that the teaching of Jesus is new indeed, compared to the Law of Moses.