What is the Obsolete Covenant? By calling this covenant “new,” he has made the first one obsolete;
  and what is obsolete and outdated will soon disappear. (Hebrews 8:13,
  NIV)


In the light of the Epistle to Hebrews, what did the author mean by "The First Covenant" and "The New Covenant"?

What is the First Covenant which is now outdated and will soon disappear?
This is the old covenant which God made with Abraham and his posterity.
Genesis 15:18 NRKJV

18 In the same day יהוה made a covenant with Abram, saying, Unto thy seed have I given this land, from the river of Egypt unto the great river, the river Euphrates:

Genesis 17:1 through 14 NRKJV

1 And when Abram was ninety years old and nine, יהוה appeared to Abram, and said unto him, I am El-Shaddai1 ; walk before me, and be thou perfect.
2 And I will make my covenant between me and thee, and will multiply thee exceedingly.
3 And Abram fell on his face: and Elohim talked with him, saying,
4 As for me, behold, my covenant is with thee, and thou shalt be a father of many nations.
5 Neither shall thy name any more be called Abram, but thy name shall be Abraham; for a father of many nations have I made thee.
6 And I will make thee exceeding fruitful, and I will make nations of thee, and kings shall come out of thee.
7 And I will establish my covenant between me and thee and thy seed after thee in their generations for an everlasting covenant, to be an Elohim unto thee, and to thy seed after thee.
8 And I will give unto thee, and to thy seed after thee, the land wherein thou art a stranger, all the land of Canaan, for an everlasting possession; and I will be their Elohim.
9 And Elohim said unto Abraham, Thou shalt keep my covenant therefore, thou, and thy seed after thee in their generations.
10 This is my covenant, which ye shall keep, between me and you and thy seed after thee; Every man child among you shall be circumcised.
11 And ye shall circumcise the flesh of your foreskin; and it shall be a token of the covenant betwixt me and you.
12 And he that is eight days old shall be circumcised among you, every man child in your generations, he that is born in the house, or bought with money of any stranger, which is not of thy seed.
13 He that is born in thy house, and he that is bought with thy money, must needs be circumcised: and my covenant shall be in your flesh for an everlasting covenant.
14 And the uncircumcised man child whose flesh of his foreskin is not circumcised, that soul shall be cut off from his people; he hath broken my covenant.

This is the New covenant which Jesus made with all people and not just the seed of Abraham.
Mat 26:27 through 29 NRKJV

27 And he took the cup, and gave thanks, and gave it to them, saying, Drink ye all of it;
28 For this is my blood of the new testament, which is shed for many for the remission of sins.
29 But I say unto you, I will not drink henceforth of this fruit of the vine, until that day when I drink it new with you in my Father's kingdom.

Since the first covenant was only with Abraham and his seed, since Jesus death would now provide salvation for both Jew and Gentile it did not cancel the old covenant, but actually expanded the first covenant to include the Gentiles.
It is called the new covenant because it is the first covenant with the Gentiles.