The section of Leviticus featured in today's reading contains the food laws for Israel (Leviticus 11). These laws are the basis of the diet for observant Jews even to today, and are very similar to the laws followed by Muslims. These food laws undoubtedly had many purposes, but one of the primary purposes of these laws was to clearly distinguish Israel from other nations.
In the NT period, issues regarding food were a huge practical problem for the early Christians. On the one hand, there were Jewish Christians who found eating with Gentiles to be an idea difficult to swallow (hardee-har-har!). This was due to the concern over eating unclean food, combined with the general distaste for associating with Gentiles. In Acts 10, God had to give Peter (who claimed never to have "eaten anything that is common or unclean" - v. 14) three special visions to convince him that God had made all foods clean (as Jesus taught in Mark 7:19). Later, when news of Peter's work with Cornelius and his household became known in Jerusalem, some Jews condemned him because he "went to uncircumcised men and ate with them" (Acts 11:3).
One of my best friends is a former Muslim, and after his conversion he eventually began to eat pork. When he later contracted a skin condition, his parents were sure it was due to his diet, and said something along the lines of "no telling what he is eating hanging around with all those white people!" Undoubtedly that is what some Jews must have thought about Peter and his association with Gentiles.
On the other hand, Gentiles had to wrestle with the practices of their background, in which pagan temples often hosted meals. Additionally, much of the meat sold at market was purchased from temples. To what extent could they participate in those meals, or purchase that meat? In 1 Corinthians 8-10 Paul explains that while they could eat meat purchased from a temple at home, they were not to do so if it might cause someone to stumble, and under no circumstances were they to participate in functions at the temples.
In the old covenant, food kept people apart. But under the new covenant, food no longer separates the clean from the unclean. And in the one bread and one cup of the Lord's supper, we can be one body.