What is the Christian definition of "the Messiah"?
The English word "Messiah" (capitalized or not) is a loanword based on a loose transliteration of the Hebrew word מָשִׁיחַ (mashiach).1 This word means "anointed," and when used as a substantive, it means "anointed one."2 "To "anoint" is "to smear oil upon."
In Mishneh Torah, Moses ben Maimon wrote,3
Throughout the generations, no one is anointed with it (the special anointing oil) except the high priests, the priest of war, and kings of the house of David alone. Even a high priest who is the son of a high priest is anointed with it, as it is said (Lev. 21:10), "And the one who is high priest among his brothers, upon whose head the anointing oil was poured..."

How is the priest anointed? The oil is poured upon his head and applied between his eyes in the form of the Greek letter chi (Χ) as it is said (Lev. 8:12), "And he poured the anointing oil on Aaron's head and anointed him to sanctify him." And the kings of the house of David are anointed [with the oil spread] like a kind of crown on their head. And they should not be anointed on other places [on their bodies], nor should one use an excessive amount of oil.

Certainly, there are many individuals referred to as "anointed" in the Tanakh, whether prophets, kings, or priests. However, we believe that Jesus is the anointed one par excellence because his anointing was greater than all other individuals who were anointed. Hence, we refer to him as the Messiah rather than a messiah. Why so?
In Psa. 45:6-7,4 it is written,
6 O' God, your throne is forever and ever. The scepter of your kingdom is a scepter of equity. You loved righteousness and hated wickedness. 7 Therefore, O' God, your God anointed you with an oil of gladness more than your brothers.
ו כִּסְאֲךָ אֱלֹהִים עוֹלָם וָעֶד שֵׁבֶט מִישֹׁר שֵׁבֶט מַלְכוּתֶךָ
ז אָהַבְתָּ צֶּדֶק וַתִּשְׂנָא רֶשַׁע עַל כֵּן מְשָׁחֲךָ אֱלֹהִים אֱלֹהֶיךָ שֶׁמֶן שָׂשׂוֹן מֵחֲבֵרֶךָ
Notice we have God (the Son) being anointed by God (the Father) with the "oil of gladness" more than his brothers. The oil of gladness is the Holy Spirit (cp. Acts 10:38).
- The Messiah is God.
- He is anointed by God the Father.
- God the Father anoints him with the Holy Spirit.
- He is anointed more than his brothers (Jesus has the Holy Spirit without measure; cp. John 3:34).
In On the City of God (De Civitate Dei), Augustine wrote,
Who is there, no matter how slow, but must here recognize Christ whom we preach, and in whom we believe, if he hears that He is God, whose throne is for ever and ever, and that He is anointed by God, as God indeed anoints, not with a visible, but with a spiritual and intelligible chrism? For who is so untaught in this religion, or so deaf to its far and wide spread fame, as not to know that Christ is named from this chrism, that is, from this anointing?
Quis non hic Christum, quem praedicamus et in quem credimus, quamlibet sit tardus, agnoscat, cum audiat Deum, cuius sedes est in saecula saeculorum, et unctum a Deo, utique sicut unguit Deus, non visibili, sed spiritali atque intellegibili chrismate? Quis enim tam rudis est in hac religione vel tam surdus adversus eius famam longe lateque diffusam, ut Christum a chrismate, hoc est ab unctione appellatum esse non noverit?
The original poster asked in a comment,
But what is the (1st coming) job according to Christians?
According to those who experienced the crucifixion and death of the Lord Jesus Christ, the earliest Christians believed that the King Messiah (i.e., the King of Israel) would be the one who "redeemed Israel" (Luke 24:21), that is, from the bondage of Gentile nations, especially Rome. However, not long thereafter, the Christians realized that the Messiah would be a redeemer, but not in the physical sense. Instead, he would liberate or redeem man from being enslaved to his sins, for the Lord Jesus Christ said (John 18:36),
My kingdom is not of this world. If my kingdom were of this world, then my servants would fight, so that I should not be delivered to the Jews. But now my kingdom is not hence.
Likewise, in Netzach Yisra'el (נצח ישראל), Ch. 42, the MaHaRaL (Judah Loew ben Bezalel) wrote,
How shall the Messiah be in this world, which is a carnal world, when all the business of the Messiah is divine, not carnal?

Footnotes
1 To be more precise, the English word "Messiah" is a Latin loanword derived from the Latin word messias, which is a transliteration of the Greek word μεσσίας (cp. John 1:41), which is a transliteration of the Hebrew word מָשִׁיחַ.
2 HALOT, p. 603; Gesenius, p. 515
3 Sefer Avoda, Hilkhot Klei ka-Mikdash veha-Ovdim Bo, Chapter 1, Halakha 7 & 9
4 Psa. 45:7-8 according to the Masoretic text.
5 Book 17, Ch. 16
References
Augustine (Aurelius Augustinus Hipponensis). On the City of God (De Civitate Dei).
Brown, Francis; Driver, Samuel Rolles; Briggs, Charles Augustus. A Hebrew and English Lexicon of the Old Testament. Oxford: Clarendon, 1906.
Gesenius, Heinrich Friedrich Wilhelm. Gesenius’s Hebrew and Chaldee Lexicon to the Old Testament Scriptures. Trans. Tregelles, Samuel Prideaux. London: Bagster, 1857.
Judah Loew ben Bezalel. Netzach Yisra'el (נצח ישראל). Jerusalem: 1964.
Moses ben Maimon. Mishneh Torah (מִשְׁנֵה תּוֹרָה). Ed. Mechon-Mamre. Jerusalem: Mechon-Mamre, 2015.