It's Not About the Animals
The best way to summarize today's passage is that Peter has a vision about animals that isn't really about animals. Yes, the dream actually has animals in it, but that's not really the point.
Peter is on the roof of a house, getting hungry, when he receives a vision. In this vision are all kinds of animals - clean and unclean. Jews were forbidden in various passages in the Old Testament from eating certain kinds of animals (called "unclean" or, using the meaning of the Greek word in this particular passage, "common") in order to set them apart from the other nations that were not called out by God (see Leviticus 20:24b–26 for this connection). A voice (presumably God, since Peter responds calling the voice "Lord") tells Peter to kill and eat. Peter, being the good Jewish boy he is, refuses. It really pictures how ingrained these traditions were in the Jewish people that Peter would tell God no when ordered - perhaps he felt that God was testing him. God replies that Peter shouldn't consider things unclean that God has called clean. Repeat 3 times. End dream sequence.
Weird. And definitely about animals, right? I think it IS significant that God specifically declares these things clean (see also statements by Jesus in Mark 7:19 and surrounding context for further proof that this is true), but as I said before, it doesn't stop there.
So, what happens next? Verse 17 tells us that Peter is perplexed at what the vision might mean (it seemed pretty straightforward to me), when three men come to the door. They want Peter to come the Cornelius, a centurion and gentile. Peter, prompted again by God, agrees and goes with them. He meets with Cornelius and in verse 28 we learn the true, deeper purpose of Peter's vision:
"You yourselves know how unlawful it is for a Jew to associate with or to visit anyone of another nation, but God has shown me that I should not call any person common or unclean."
It's not really about the animals at all. The Spirit of God has been moving through the world through His people. The Gospel has spread from Jerusalem to Judea, Judea to Samaria. But up to this point those that have received the Good News of Jesus have been Jews. This is a vision about people. This is a vision about the Gospel going to EVERYONE, or, as Acts 1:8 says, "to the ends of the earth". This is more than just about Jews being able to change their diet. It is about the Truth of God for ALL people.