How was a covenant sealed in the Old Testament? Between two different tribes or God and Abraham?

This is an interesting question. I wish I knew just a little more of what you are thinking of. However, I will make some observations and hope that one or two of them will help you in your study.

A quick search shows only one verse in the Bible that actually refers to the sealing of a covenant using those words. The more common word used to refer to the ratification of a covenant is the word confirm. At least five scriptures speak of the confirming of covenants (1 Chronicles 16:17And hath confirmed the same to Jacob for a law, and to Israel for an everlasting covenant,
See All...; Psalm 105:10And confirmed the same unto Jacob for a law, and to Israel for an everlasting covenant:
See All...; Daniel 9:27And he shall confirm the covenant with many for one week: and in the midst of the week he shall cause the sacrifice and the oblation to cease, and for the overspreading of abominations he shall make it desolate, even until the consummation, and that determined shall be poured upon the desolate.
See All...; Galatians 3:15Brethren, 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.
See All..., 17). Let's look at the different elements separately.

As you evidently know, a covenant is an agreement between two parties. Covenants can be made between two different men or entities (as nations) or between God and man. Covenants between God and man can take two different forms. They can be conditional. That is, God will promise to do a certain thing if man will do his part. The law of Moses was a conditional covenant. Or, they can be unconditional. That is, God will do His part no matter how faithful or unfaithful man is. The promise to Abraham in Genesis 15 is an unconditional covenant.

Nehemiah 9:38And because of all this we make a sure covenant, and write it; and our princes, Levites, and priests, seal unto it.
See All... states, "And because of all this we make a sure covenant, and write it; and our princes, Levites, and priests, seal unto it." As I said, this is the only verse where it specifically states that a covenant is sealed (where the words covenant and seal are used). It is an agreement among the Israelites who have returned from Babylon to Judah to maintain their separation from the peoples around them. This covenant took the form of a document. In cases like this, the sealing was often done by closing the document (sometimes in a scroll), putting a seal of wax or clay on the document, and imprinting it with a metal seal (sometimes on a ring). The document could not be opened without breaking the seal. I do not know that this was physically done in this case. But the following scriptures will show cases where it was done this way.

1 Kings 21:8So she wrote letters in Ahab's name, and sealed them with his seal, and sent the letters unto the elders and to the nobles that were in his city, dwelling with Naboth.
See All... So she wrote letters in Ahab's name, and sealed them with his seal, and sent the letters unto the elders and to the nobles that were in his city, dwelling with Naboth.

Esther 3:12Then were the king's scribes called on the thirteenth day of the first month, and there was written according to all that Haman had commanded unto the king's lieutenants, and to the governors that were over every province, and to the rulers of every people of every province according to the writing thereof, and to every people after their language; in the name of king Ahasuerus was it written, and sealed with the king's ring.
See All... Then were the king's scribes called on the thirteenth day of the first month, and there was written according to all that Haman had commanded unto the king's lieutenants, and to the governors that were over every province, and to the rulers of every people of every province according to the writing thereof, and to every people after their language; in the name of king Ahasuerus was it written, and sealed with the king's ring.

Jeremiah 32:10-11 [10] And I subscribed the evidence, and sealed it, and took witnesses, and weighed him the money in the balances. [11] So I took the evidence of the purchase, both that which was sealed according to the law and custom, and that which was open:
See All... And I subscribed the evidence, and sealed it, and took witnesses, and weighed him the money in the balances. So I took the evidence of the purchase, both that which was sealed according to the law and custom, and that which was open:

However, I suspect that you are referring more directly to the confirmation of a covenant. As I consider the covenants of the Bible, I do see that they were not all confirmed in the same way. For instance, the covenant of the law seems to be confirmed in Exodus 24 (although neither words, confirm or seal, are used here).

Exodus 24:7-8 [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.
See All... 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. 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.

Moses read the book to them. They verbally agreed to keep the covenant. (Compare this with Deuteronomy 27:26Cursed be he that confirmeth not all the words of this law to do them. And all the people shall say, Amen.
See All... - "Cursed be he that confirmeth not all the words of this law to do them. And all the people shall say, Amen.") Then, Moses took blood and sprinkled it on the people and declared the covenant made.

In other instances, people confirmed covenants in interesting ways. Ruth 4:7Now this was the manner in former time in Israel concerning redeeming and concerning changing, for to confirm all things; a man plucked off his shoe, and gave it to his neighbour: and this was a testimony in Israel.
See All... states, "Now this was the manner in former time in Israel concerning redeeming and concerning changing, for to confirm all things; a man plucked off his shoe, and gave it to his neighbour: and this was a testimony in Israel." So, the removal of the shoe and handing it to his neighbor was the confirmation of the covenant.

However, your specific mention of the covenant between Abraham and God makes me believe that you are thinking of Genesis 15. This is a powerful confirmation and evidently followed the pattern men would take in covenants with other men--with one major exception. Let us go first to another passage:

Jeremiah 34:18-19 [18] And I will give the men that have transgressed my covenant, which have not performed the words of the covenant which they had made before me, when they cut the calf in twain, and passed between the parts thereof, [19] The princes of Judah, and the princes of Jerusalem, the eunuchs, and the priests, and all the people of the land, which passed between the parts of the calf;
See All... And I will give the men that have transgressed my covenant, which have not performed the words of the covenant which they had made before me, when they cut the calf in twain, and passed between the parts thereof, The princes of Judah, and the princes of Jerusalem, the eunuchs, and the priests, and all the people of the land, which passed between the parts of the calf;

This passage refers to one specific practice of Bible times. In order to confirm a covenant, two parties would kill a calf and cut it into two pieces. Then, they would pass together between the two parts of the dead animal as a solemn sign of both parties that they fully agreed to the terms of the covenant. They confirmed or sealed the covenant by passing through the parts of the calf.

A similar act takes place in Genesis 15. But it is also very different. Look carefully at these verses (or go and read the entire passage):

Genesis 15:9-10 [9] And he said unto him, Take me an heifer of three years old, and a she goat of three years old, and a ram of three years old, and a turtledove, and a young pigeon. [10] And 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.
See All... And he said unto him, Take me an heifer of three years old, and a she goat of three years old, and a ram of three years old, and a turtledove, and a young pigeon. And 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.

Genesis 15:12And when the sun was going down, a deep sleep fell upon Abram; and, lo, an horror of great darkness fell upon him.
See All... And when the sun was going down, a deep sleep fell upon Abram; and, lo, an horror of great darkness fell upon him. Genesis 15:17-18 [17] And it came to pass, that, when the sun went down, and it was dark, behold a smoking furnace, and a burning lamp that passed between those pieces. [18] In the same day the LORD 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:
See All... And it came to pass, that, when the sun went down, and it was dark, behold a smoking furnace, and a burning lamp that passed between those pieces. In the same day the LORD 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:

In this instance, five different animals are killed (the same five that are commonly used in the sacrifices described in Leviticus). Then, the covenant is confirmed after God had placed Abram (Abraham) into a deep sleep. God, in the form of a smoking furnace and a burning lamp, passed between the pieces by Himself. He does not take Abraham through with Him. I believe this means that this is an unconditional covenant and God alone is responsible for the fulfillment of this covenant. Even if Abraham and his seed fail, God will still bring the covenant to pass. It would be good for us to know the content of this covenant.