Messianic Jews are those who believe that Yeshua (Jesus) is the Messiah.
Obviously, many of the original believers, including all of the 12 disciples, were Jews. Many Jews do still believe although most Jews have barely heard about their Jewish brothers and sisters believing in Yeshua.
See the Susya page for videos about a 3-4th century synagogue that includes a mosaic inscription to Yeshua.
First Messianic Jewish Synagogue Found? (Magdala) (Israel Today)