Would you please explain what the Ark is in 1 Kings 8 that was brought to the Temple for King Solomon.

There are three arks in the Bible. Each one is a rectangular shaped container with a cover. It might help if we look at each one separately.

NOAH'S ARK:

This one, of course, is easy. This is the ark of gopher wood that the Lord commanded Noah to build as a boat so that his family and the land animals might be preserved during the judgment of the flood (Genesis 6:14-19 [14] Make thee an ark of gopher wood; rooms shalt thou make in the ark, and shalt pitch it within and without with pitch. [15] And this is the fashion which thou shalt make it of: The length of the ark shall be three hundred cubits, the breadth of it fifty cubits, and the height of it thirty cubits. [16] A window shalt thou make to the ark, and in a cubit shalt thou finish it above; and the door of the ark shalt thou set in the side thereof; with lower, second, and third stories shalt thou make it. [17] And, behold, I, even I, do bring a flood of waters upon the earth, to destroy all flesh, wherein is the breath of life, from under heaven; and every thing that is in the earth shall die. [18] But with thee will I establish my covenant; and thou shalt come into the ark, thou, and thy sons, and thy wife, and thy sons' wives with thee. [19] And of every living thing of all flesh, two of every sort shalt thou bring into the ark, to keep them alive with thee; they shall be male and female.
See All...). Though it was large (300 cubits or at least 450 feet long), it matches the definition of an ark given above. It was a rectangular box-shaped boat with a cover.

MOSES' ARK OF BULRUSHES:

The ark of bulrushes was certainly tiny compared to the ark of Noah. It only had to carry one baby boy. However, it did float on water like Noah's ark (in this case, the Nile River). It was made of bulrushes and was used to preserve Moses from the command of the Pharaoh of Egypt to kill all the Hebrew baby boys (Exodus 2:1-4 [1] And there went a man of the house of Levi, and took to wife a daughter of Levi. [2] And the woman conceived, and bare a son: and when she saw him that he was a goodly child, she hid him three months. [3] And when she could not longer hide him, she took for him an ark of bulrushes, and daubed it with slime and with pitch, and put the child therein; and she laid it in the flags by the river's brink. [4] And his sister stood afar off, to wit what would be done to him.
See All...). We do not know its dimensions, but we can be confident that it had a rectangular box-shape seeing this is the shape that would hold a baby. We also know that it had a cover because when Pharaoh's daughter found this ark floating in the river, she "opened it" in order to see the baby (Exodus 2:5-6 [5] And the daughter of Pharaoh came down to wash herself at the river; and her maidens walked along by the river's side; and when she saw the ark among the flags, she sent her maid to fetch it. [6] And when she had opened it, she saw the child: and, behold, the babe wept. And she had compassion on him, and said, This is one of the Hebrews' children.
See All...). However, this is not the ark that Solomon brought to the Temple.

ARK OF THE COVENANT:

The third ark was made of what was called shittim wood and was built at the commandment of the Lord as the most important piece of furniture in the tabernacle. Moses rode in the ark of bulrushes, but he had the ark of the covenant made at the commandment of God (Exodus 25:10-16 [10] And they shall make an ark of shittim wood: two cubits and a half shall be the length thereof, and a cubit and a half the breadth thereof, and a cubit and a half the height thereof. [11] And thou shalt overlay it with pure gold, within and without shalt thou overlay it, and shalt make upon it a crown of gold round about. [12] And thou shalt cast four rings of gold for it, and put them in the four corners thereof; and two rings shall be in the one side of it, and two rings in the other side of it. [13] And thou shalt make staves of shittim wood, and overlay them with gold. [14] And thou shalt put the staves into the rings by the sides of the ark, that the ark may be borne with them. [15] The staves shall be in the rings of the ark: they shall not be taken from it. [16] And thou shalt put into the ark the testimony which I shall give thee.
See All...). This ark was also shaped like a rectangular box. It's dimensions in cubits were 2 1/2 long by 1 1/2 wide by 1 1/2 high. That is approximately 3 feet and 9 inches long and 2 feet and 3 inches wide and high. It was actually quite small. It also had a covering on it. This significant covering is called the "mercy seat" (Exodus 26:34And thou shalt put the mercy seat upon the ark of the testimony in the most holy place.
See All...).

The ark of the covenant was hollow inside like a box and at times held various items which God commanded to be placed there. At one time it carried "the golden pot that had manna, and Aaron's rod that budded, and the tables of the covenant" (Hebrews 9:4Which had the golden censer, and the ark of the covenant overlaid round about with gold, wherein was the golden pot that had manna, and Aaron's rod that budded, and the tables of the covenant;
See All...). By the time Solomon moved it to the newly built Temple, it only had the two tables of stone on which Moses had written the Ten Commandments (1 Kings 8:9There was nothing in the ark save the two tables of stone, which Moses put there at Horeb, when the LORD made a covenant with the children of Israel, when they came out of the land of Egypt.
See All...). It was a special type of the revelation of God's truth to man. Not only did it contain the Ten Commandments, but God promised Moses that "there I will meet with thee, and I will commune with thee from above the mercy seat" (Exodus 25:22And there I will meet with thee, and I will commune with thee from above the mercy seat, from between the two cherubims which are upon the ark of the testimony, of all things which I will give thee in commandment unto the children of Israel.
See All...). This ark that was built by Moses was the ark that was moved into Solomon's Temple.

APPLICATION:

Finally, I would like to mention the spiritual picture given by the three arks. Noah's ark is a picture of salvation. It is deliverance from the wrath of God. The ark of bulrushes is a picture of God's protection. It is deliverance from the wrath of man. The ark of the covenant is a picture of communion and fellowship. It is the promise that God will meet with us and commune with us if we seek Him. Each person needs each of these arks at some time in their life. I thank the Lord for each of them.