Messiah and Messianic Exegesis
Messiah means "anointed one" (Hebrew mashiach, transliterated into Greek as messias or translated as christos). Although the Old Testament can use the term for priests and prophets (Exodus 28:41, 29:7, 1 Kings 19:16), Messiah refers to the coming "king" from the tribe of Judah (see Genesis 49:10) and the lineage of David, who would restore and unite Israel, vindicate God's people, condemn evil doers and overthrow the forces of evil, enable God's people to understand and obey Torah, perhaps draw the gentiles to Yhwh, and establish edenic peace in the cosmos (beginning with Zion, "this holy mountain").
The concept of Messiah developed over centuries of experience and theological reflection. It is difficult to pinpoint the specific biblical texts that explicitly promise this coming king, and opinion in the Judaism of Jesus' day was mixed about the messianic hope. A group of biblical scholars today are even challenging the existence of a widespread messianic expectation in ancient Jewish thought (e.g., John J. Collins). One can classify this school of thought as minimalist.
Your professor is a maximalist, who traces the development of the Messiah concept from the promise of a son to king David who would build a temple for Yhwh and establish an eternal (Heb., le-'olam) reign (2 Samuel 7:12-16). The second step of this development is in the visions of the Book of Isaiah, especially the coming "shoot of Jesse" (i.e., David's father Jesse), upon whom the Spirit rested, who would bring righteousness and establish shalom, the peace of Eden (Isaiah 11:1-10). The third step is to be found in the the exilic prophet Ezekiel, who envisions a future shepherd David, who would unite the broken and scattered tribes of Israel, bring an internal change in God's people, and establish a covenant of shalom (Ezekiel 37:16-28). A fourth step is in the apocalyptic tradition of Daniel, which portrays an otherworldly "stone" that would crush the present evil human kingdom(s) and God establishing an eternal kingdom in its place (Dan 2:34-35, 44). Moreover, Daniel sees a "son of man" who comes on the clouds and is given an eternal reign (Dan 7:13-14).
To this list can be added many texts that speak of a coming king (e.g., Micah 5:2, Zechariah 9:9-10). Your professor argues that these "messianic texts" can be identified by three features:
Visionary texts that contain these three features would have been viewed by Jews of Jesus' day as messianic. Not every text that NT authors identifiy as "fulfilled" by Jesus meet these qualifications, so other Jews would not necessarily accept them.