Isaiah 11:1 promises a descendant of Jesse, a son of David, who will have the Spirit of the LORD resting on him. That is what it means to be anointed and that is what is promised of the Messiah.
Messiah, in Hebrew, means "anointed" and does not immediately point to only one single person. In a sense every king of Judah, from Saul onwards, had been the messiah. Both priests and kings were anointed, but "the LORD's anointed" was shorthand for "the King" (1 Sam 24:10, Lam 4:20).
There is one aspect of this that points to Jesus particularly. He is the anointed King and the anointed High Priest. He is particularly anointed to fulfill both roles.