First, what DocCathode said. Although there is considerable discussion and dispute about exactly what will happen when the Messiah comes, there are some things that are pretty much generally accepted in traditional Judaism. When the Messiah comes, God’s Kingdom on Earth will be established, with Jerusalem as its center. The dead will be restored to life. There will be peace on earth, the lion will lie down with the lamb. None of that happened, hence the Messiah hasn’t come yet.
Now, the Christians say all those things will happen when Jesus returns (the Second Coming.) Jews say, OK, that’s possible, and when (if) it happens, we’ll accept him as the Messiah.
Now, second level is that we reject the idea of Jesus as God-incarnate. To Jews, the concept of a god being born from a human woman impregnated by a god is… well, pagan, if not downright silly. Greek myths abound with such stories. Jews reject the notion of God becoming human or impregnating a human woman.
And Jews certainly reject the notion of a trinity: monotheism is the core of Judaism. Christians argue that the trinity doesn’t conflict with monotheism, but (speaking frankly) Jews don’t buy that rationalization.
Finally, Jews believe that we have a covenant with God. We were told God’s laws, His rules of conduct, how to behave, how to repent and seek forgiveness when we err. And that covenant was eternal, for all generations. Christianity attempts to overthrow those rules, replace them with a new process (belief in Jesus instead of behaviors.) We don’t believe that God lied to us; hence, we reject the basic principle of Christianity.
So, in summary, Jews generally think:
(1) Jesus has not fulfilled Messianic prophecy
(2) the concept of God-incarnate is inconsistent with the nature of God
(3) the concept of a trinity is inconsistent with monotheism
(4) the notion that God would lie about the covenant is contrary to the nature of God.