Concise Bible Dictionary:

To this subject as spoken of in scripture there are two branches:
1. Man’s covenant with his fellow, or nation with nation, in which the terms are mutually considered and agreed to: it is then ratified by an oath, or by some token, before witnesses. Such a covenant is alluded to in Galatians 3:1515Brethren, I speak after the manner of men; Though it be but a man's covenant, yet if it be confirmed, no man disannulleth, or addeth thereto. (Galatians 3:15): if a man’s covenant be confirmed it cannot be disannulled or added to. When Abraham bought the field of Ephron in Machpelah, he paid the money “in the audience of the sons of Heth” as witnesses, and it was thus made sure unto him (Gen. 23:1616And Abraham hearkened unto Ephron; and Abraham weighed to Ephron the silver, which he had named in the audience of the sons of Heth, four hundred shekels of silver, current money with the merchant. (Genesis 23:16)). In the covenant Jacob made with Laban, they gathered a heap of stones to be witness between them, and “they did eat there upon the heap” (Gen. 31:4646And Jacob said unto his brethren, Gather stones; and they took stones, and made an heap: and they did eat there upon the heap. (Genesis 31:46)). When the Gibeonites deceived Joshua and the heads of Israel, “the men took of their victuals, and asked not counsel at the mouth of the Lord, and... sware unto them” (Josh. 9:14-1514And the men took of their victuals, and asked not counsel at the mouth of the Lord. 15And Joshua made peace with them, and made a league with them, to let them live: and the princes of the congregation sware unto them. (Joshua 9:14‑15)). So to this day, if a stranger in the East can get the head of a tribe to eat with him, he knows he is safe, the eating is regarded as a covenant. In 2 Chronicles 13:55Ought ye not to know that the Lord God of Israel gave the kingdom over Israel to David for ever, even to him and to his sons by a covenant of salt? (2 Chronicles 13:5) we read of “a covenant of salt”; and to eat salt together is also now regarded as a bond in the East.
2. The covenants made by God are of a different order. He makes His covenants from Himself, without consulting man. With Noah God made a covenant that he would not again destroy the world by a flood, and as a token of that covenant, He set the rainbow in the cloud (Gen. 9:8-178And God spake unto Noah, and to his sons with him, saying, 9And I, behold, I establish my covenant with you, and with your seed after you; 10And with every living creature that is with you, of the fowl, of the cattle, and of every beast of the earth with you; from all that go out of the ark, to every beast of the earth. 11And I will establish my covenant with you; neither shall all flesh be cut off any more by the waters of a flood; neither shall there any more be a flood to destroy the earth. 12And God said, This is the token of the covenant which I make between me and you and every living creature that is with you, for perpetual generations: 13I do set my bow in the cloud, and it shall be for a token of a covenant between me and the earth. 14And it shall come to pass, when I bring a cloud over the earth, that the bow shall be seen in the cloud: 15And I will remember my covenant, which is between me and you and every living creature of all flesh; and the waters shall no more become a flood to destroy all flesh. 16And the bow shall be in the cloud; and I will look upon it, that I may remember the everlasting covenant between God and every living creature of all flesh that is upon the earth. 17And God said unto Noah, This is the token of the covenant, which I have established between me and all flesh that is upon the earth. (Genesis 9:8‑17)). This kind of covenant takes the form of an unconditional promise. Such was God’s covenant with Abraham, first as to his natural posterity (Gen. 15:4-64And, behold, the word of the Lord came unto him, saying, This shall not be thine heir; but he that shall come forth out of thine own bowels shall be thine heir. 5And he brought him forth abroad, and said, Look now toward heaven, and tell the stars, if thou be able to number them: and he said unto him, So shall thy seed be. 6And he believed in the Lord; and he counted it to him for righteousness. (Genesis 15:4‑6)); and secondly, as to his seed, Christ (Gen. 22:15-1815And the angel of the Lord called unto Abraham out of heaven the second time, 16And said, By myself have I sworn, saith the Lord, for because thou hast done this thing, and hast not withheld thy son, thine only son: 17That in blessing I will bless thee, and in multiplying I will multiply thy seed as the stars of the heaven, and as the sand which is upon the sea shore; and thy seed shall possess the gate of his enemies; 18And in thy seed shall all the nations of the earth be blessed; because thou hast obeyed my voice. (Genesis 22:15‑18)). He gave him also the covenant of circumcision (Gen. 17:10-1410This is my covenant, which ye shall keep, between me and you and thy seed after thee; Every man child among you shall be circumcised. 11And ye shall circumcise the flesh of your foreskin; and it shall be a token of the covenant betwixt me and you. 12And 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. 13He 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. 14And 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. (Genesis 17:10‑14); Acts 7:88And he gave him the covenant of circumcision: and so Abraham begat Isaac, and circumcised him the eighth day; and Isaac begat Jacob; and Jacob begat the twelve patriarchs. (Acts 7:8))—a seal of the righteousness of faith (Rom. 4:1111And he received the sign of circumcision, a seal of the righteousness of the faith which he had yet being uncircumcised: that he might be the father of all them that believe, though they be not circumcised; that righteousness might be imputed unto them also: (Romans 4:11)).
The covenant with the children of Israel at Sinai, on the other hand, was conditional: if they were obedient and kept the law they would be blessed; but if disobedient they would be cursed (Deut. 27-28).
In the Epistle to the Galatians the apostle argues that the “promise” made by God—”the covenant that was confirmed before of God in Christ” could not be affected by the law which was given 430 years later (Gal. 3:16-1716Now to Abraham and his seed were the promises made. He saith not, And to seeds, as of many; but as of one, And to thy seed, which is Christ. 17And this I say, that the covenant, that was confirmed before of God in Christ, the law, which was four hundred and thirty years after, cannot disannul, that it should make the promise of none effect. (Galatians 3:16‑17)). The promise being through Christ, the apostle could add respecting Gentile believers, “If ye be Christ’s, then are ye Abraham’s seed, and heirs according to promise” (Gal. 3:2929And if ye be Christ's, then are ye Abraham's seed, and heirs according to the promise. (Galatians 3:29)).

From Manners and Customs of the Bible:

This was a very ancient method of making a covenant. The two contracting parties slaughtered a victim, cut the body in two, and passed between the severed parts. Some writers hold that the design was to express a wish that, if the covenant should be broken, the same fate might befall the party violating it which had befallen the slain beast. Others think that it was intended to represent, that as the two divided parts belonged to one animal, so the two parties making the covenant were of one mind so far as the subject of the covenant was concerned. It is thought probable that the latter was the original design of the custom, and that the former notion was added to the meaning subsequently, or substituted for it when the original intention was forgotten. This old custom is referred to in the very expression which was used by the Hebrews to represent the making of a covenant. The phrase “make a covenant,” which is so often used in the Old Testament, is literally, “to cut a covenant” (karath berith).
This ceremony was used when Jehovah made a covenant with Abram. See Genesis 15:10-1110And he took unto him all these, and divided them in the midst, and laid each piece one against another: but the birds divided he not. 11And when the fowls came down upon the carcases, Abram drove them away. (Genesis 15:10‑11). “Ephraem Syrus observes, that God condescended to follow the custom of the Chaldeans, that he might in the most solemn manner confirm his oath to Abram the Chaldean” (Keil and Delitzsch, Commentary in loco). The custom was widespread among ancient nations, and is often referred to by classical writers. There are traces of it even in modern times. Pitts, after narrating some of the superstitious customs of the Algerine pirates when a storm overtakes them at sea, continues: “If they find no succor from their before—mentionedrites and superstitions, but that the danger rather increases, then they go to sacrificing of a sheep, (or two or three upon occasion, as they think needful,) which is done after this manner: having cut off the head with a knife, they immediately take out the entrails, and throw them and the head overboard; and then with all the tipped they can (without skinning) they cut the body into two parts by the middle, and throw one part over the right side of the ship and the other over the left, into the sea, as a kind of propitiation” (Religion and Manners of the Mahometans, chap. 2).

Ministry Nuggets:

 Covenant, when used in connection with the Lord, is always, it seems to me, some order established by God and announced to man, according to the terms of which He enters into relationship with man, or according to which man is to approach Him. (Genesis 9 by J.N. Darby)