An Ethnically-Patrilineal and/or -Matrilineal Jew who believes that the historical Jesus of Nazareth (Yeshua HaNotzri) is the Messiah and that the New Testament is a part of TaNaKH(i.e., Torah, Nevi'im, Ketuvim, Hadashah-- literally "New" or "News").
A Messianic Jew does not necessarily keep all 613 mitzvot in the Torah, may not hold the Talmud as part of the Jewish Bible, and may be a member of a church as opposed to a Messianic shul or synagogue. Also called "Jews for Jesus", Messianic Jews are often targeted for persecution by Antimissionaries such as the Antimissionary organization Jews for Judaism.
Most of the time, it is a Christian that engages in attempting to practice Jewish rituals and traditions then parading themselves around as a "Messianic Jew" or "Jewish Christian." They are not in fact Jewish nor recognized as Jews by any branch of mainstream Judaism. Additionally, they are lambasted for using this misleading title by trying to draw true Jews astray from Judaism when they proselytize.
Person #1: Hey, I'm Jewish.
Person #2: Really?! Yeah, me too!
Person #1: Really, I'm Reform, what are you?
Person #2: I'm a Messianic Jew!
Person #1: Is your mother Jewish?
Person #2: Erm...well...no, but I still like go to church oops I mean synangogue and stuff on Saturday and where a watch-ma-call-it....um...yarmulke! We just believe that Yeshua (Jesus) is the moshiach (messiah), that's all!
Person #1: Um, no, Dude, you're definitely not Jewish. You're a Christian.