The word messiah comes from the Hebrew and Arabic languages and means anointed-one.
Likewise, the word christ comes from the Greek language and also means anointed-one.
Therefore, Jesus Christ (or Jesus the Christ) is the equivalent of Jesus the Messiah, both meaning "Jesus, the Anointed-One of God."
In the Hebrew Scriptures (Old Testament), there were many 'messiahs' (people anointed by God to act as judges and saviors).
However, there is also foretold of a particular messiah, with very specific qualifications, who was to be the chosen-one of God who would be the redeemer, king, and shepherd of the entire world.
This redeemer would be a descendent of David (a king who pleased God because of his repentant heart).
(Note: These are written after King David had died.)
"And I will set up over them one shepherd, my servant David..." (Ezekiel 34:23)
"A shoot shall come out from the stump of Jesse (Jesse is David's father), and a branch shall grow out of his roots." (Isaiah 11:1-2)
"Here is the man whose name is the Branch...and he will be clothed with majesty and will sit and rule on his throne." (Zechariah 6:12-13)
These terms of "David," "shoot of Jesse," and "Branch" all indicate that this one king and one shepherd will be a descendant of David.
However, there are also passages that indicate this king will rule forever, which means he is no ordinary king.
"As I watched in the night visions, I saw one like a son of man (human being) coming with the clouds of heaven. And he came to the Ancient One (God) and was presented before him.
To him was given dominion and glory and kingship, that all peoples, nations, and languages should serve him. His dominion is an everlasting dominion that shall not pass away, and his kingship is one that shall never be destroyed." (Daniel 7:13-14)
Thus, so far, we can see that the messiah of God is to be the king and shepherd of the world, who is a descendant of David, and who is like a human (son of man), and has an everlasting kingdom.
The Old Testament is also very clear that God is the only king and the only shepherd.
"Yet my eyes have seen the King, the LORD (Yahweh/God) of hosts!" (Isaiah 6:5)
"And the LORD (Yahweh/God) will become king over all the earth;" (Zechariah 14:9)
"The LORD is my shepherd, I shall not want." (Psalms 23:1)
"...by the hands of the Mighty One of Jacob, by the name of the Shepherd, the Rock of Israel, by the God of your father, who will help you, by the Almighty who will bless you..." (Genesis 49:24-25)
As is seen in these verses, the words King, Shepherd, Rock of Israel, and Almighty are all terms for God. He is the one, true, and only king and shepherd of the world.
The only way these two ideas can be combined is if this descendant of David, who is in the form of a man, is also God.
Interestingly, this idea of a God-man is actually in scripture.
"And above the dome over their heads there was something like a throne...and seated above the likeness of a throne was something that seemed like a human form.
...This was the appearance of the likeness of the glory of God (Yahweh/LORD)." (Ezekiel 1:26-28)
In addition, there are also verses that foretell a king being born who will be called God.
"For a child has been born for us, a son given to us; authority rests upon his shoulders; and he is named Wonderful Counselor, Mighty God, Everlasting Father, Prince of Peace.
His authority shall grow continually, and there shall be endless peace for the throne of David and his kingdom." (Isaiah 9:6-7)
A child called Mighty God and Everlasting Father?
This is not an ordinary child being born. This is very explicitly talking about God in the flesh.
These two verses alone, from the book of Isaiah, indicate that the messiah of God will be:
In the form of a human, a descendant of David, a king, and God.
There are a lot of other scriptures that talk about this messiah of God, other than the ones above, and they all indicate a very particular "being."
Already, as just noted, just from the scriptures above, this messiah would have to be God in the form of a man who is a descendent of David.
However, there are even more descriptions of the messiah that narrows down even his actions.
"...Shout aloud, O daughter Jerusalem! Lo, your king comes to you; triumphant and victorious is he, humble and riding on a donkey." (Zechariah 9:9)
Also, in Isaiah 53, there is described what is known as the suffering servant.
This servant is a man who does no wrong, and yet is punished.
However, in being punished, he takes on the wrongs of all other people, making them righteous (and thus making him a savior and messiah).
"But he was wounded for our transgressions (sins), crushed for our iniquities; upon him was the punishment that made us whole, and by his bruises we are healed." (Isaiah 53:5)
"He was oppressed, and he was afflicted, yet he did not open his mouth; like a lamb that is led to the slaughter, and like a sheep that before it's shearers is silent, so he did not open his mouth." (Isaiah 53:7)
"By a perversion of justice he was taken away..." (Isaiah 53:8)
"They made his grave with the wicked and his tomb with the rich, although he had done no violence, and there was no deceit in his mouth." (Isaiah 53:9)
"Yet it was the will of the LORD (Yahweh/God) to crush him with pain. When you make his life an offering for sin... through him the will of the LORD shall prosper." (Isaiah 53:10)
With all of the Old Testament (Hebrew Scriptures) in consideration, it is clear that there is such a person who has fulfilled all the qualifications of this foretold messiah; and the good news of his coming is recorded in the New Testament.
The Messiah of God and savior of the world... Jesus Christ.