The NY Times reports the death of the grand rabbi of a Hasidic Jewish sect. Without actually using the term messiah, the article notes that "[i]n Hasidism, a mystical brand of Orthodox Judaism, the grand rabbi is revered as a kinglike link to God, holding vast sway over members' lives." The Times article reminded me that many members of another Hasidic sect, the Lubavitchers, regarded their late rabbi, known simply as The Rebbe, as the Mashiach, or the Jewish Messiah. Wikipedia provides a list of other Jewish messiah claimants, as well as a very readable general article about Jewish views of the Messiah.
Recall the scene in the last part of Acts chapter 5: Peter and the other apostles proclaim to the Sanhedrin that God has raised Jesus from the dead and exalted him to his right hand as Prince and Savior. At this, the council members become furious and want to put the Apostles to death. But Gamaliel, a respected rabbi, has the Apostles taken out of the room so that he can address the Sanhedrin in executive session, so to speak. He reminds his audience that the Apostles' messianic claim about Jesus was nothing new: Others had made similar claims for themselves, and had led fatally-unsuccessful revolts against the Roman conquerors. So (Gamaliel says), let's not put these guys to death; let's just wait and see what happens to them. That cools down the Sanhedrin hotheads enough that they let the Apostles go with "just" a flogging. After that, "[d]ay after day, in the temple courts and from house to house, [the Apostles] never stopped teaching and proclaiming the good news that Jesus is the Messiah" (Acts 5.42).
Evidently, at that time the Apostles believed Jesus to be the Messiah, in the classic Jewish sense of the term: [REVISED:] The Anointed One, a king designated by God who would bring the whole world to the LORD and restore Israel to its rightful place in God's service. Jesus being dead, the Apostles added the detail that Jesus was seated at God's right hand, whence he would imminently return to carry out his Messiah duties.
(Notably, not once does Acts recount that the Apostles ever proclaimed Jesus to be God incarnate.)