Source: Sid Z. Leiman, The Canonization of Hebrew Scripture: The Talmudic and Midrashic Evidence (Hamden: Archon, 1976) 103-104.
According to the rabbis, a person who touches an impure object gains second degree impurity with respect to his hands. One removes such impurity through ritually washing them. Leiman states: “If the hands are not ritually washed, they in turn defile the priestly gifts (terumah) which they touch. Such priestly gifts may no longer be eaten and must be burned.” Eventually, unwashed hands defiled all food that they touched, not only that of priests. This may be because the rabbis were trying to make ordinary meals an experience of the divine.
This rabbinic ritual may go back to New Testament times, since the Pharisees criticized Jesus’ disciples for not washing their hands. For them, unwashed hands rendered the food defiling to the one who eats it (Matthew 15; Mark 7).
I remember reading an article from the Worldwide Church of God on clean and unclean meats. In Acts 10:14, Peter said he never ate food that was common or unclean. According to the article, the unclean meats referred to those prohibited in Leviticus 11, whereas the common ones were meats defiled through unwashed hands. That may be true, since there seems to be a distinction between common and unclean food in the verse.