A testament is a covenant that is based on something received as the result of someone or something dying (Hebrews 9:16-17). We still use the word today in the idea of a last will and testament--a document which determines who gets what after the author of the will and testament dies.
The "old testament" technically refers to the covenant of the law made with the children of Israel. In 2 Corinthians 3:14, the "old testament" refers specifically to the written law of Moses--"But their minds were blinded: for until this day remaineth the same vail untaken away in the reading of the old testament; which vail is done away in Christ." The "old testament" was read and therefore refers to the writing here.
However, the old testament is also called the "first testament." Hebrews 9:15 speaks of "the redemption of the transgressions that were under the first testament." This indicates that the first, or old, testament did not completely do away with sins but simply held them in check until the perfect sacrifice of Christ. But the first testament did require the shedding of blood. Consider this passage:
Hebrews 9:18 Whereupon neither the first testament was dedicated without blood.
19 For when Moses had spoken every precept to all the people according to the law, he took the blood of calves and of goats, with water, and scarlet wool, and hyssop, and sprinkled both the book, and all the people,
20 Saying, This is the blood of the testament which God hath enjoined unto you.
21 Moreover he sprinkled with blood both the tabernacle, and all the vessels of the ministry.
22 And almost all things are by the law purged with blood; and without shedding of blood is no remission.
This passage refers specifically back to an incident when Moses sprinkled the blood of the covenant in Exodus after the people agreed to keep the commandments of the law.
Exodus 24:7 And he took the book of the covenant, and read in the audience of the people: and they said, All that the LORD hath said will we do, and be obedient.
8 And Moses took the blood, and sprinkled it on the people, and said, Behold the blood of the covenant, which the LORD hath made with you concerning all these words.
As the old testament was based on the sacrifice of animals (Hebrews 9:18), so the new testament is based on the sacrificial death of Jesus Christ. Jesus is called "the mediator of the new testament" (Hebrews 9:15). That is why He called His shed blood (as pictured in the fruit of the vine) "my blood of the new testament" (Matthew 26:28). The new testament is the source of eternal life for those who believe in Jesus Christ. In contrast to the old testament, the new testament provides full forgiveness for sins and eternal salvation.
Hebrews 10:10 By the which will we are sanctified through the offering of the body of Jesus Christ once for all.
11 And every priest standeth daily ministering and offering oftentimes the same sacrifices, which can never take away sins:
12 But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God;
In time, the inspired books that were written before the coming of Christ came to be known as the Old Testament, while the inspired books that were written after Christ came and died on the cross were known as the New Testament. However, the phrases originally referred to two foundations for forgiveness--the sacrifice of animals and the sacrifice of Jesus Christ.