What does Matthew 12:40 mean?

Some scribes and Pharisees have asked Jesus for a sign: demanding a supernatural miracle to prove that He is truly the Messiah. This is not a sincere request, since Jesus has very recently performed miracles (Matthew 12:9–13; 22), and critics attributed them to Satan (Matthew 12:24). These men are simply rejecting what evidence they see, then demanding more, in a never-ending cycle.

Jesus has flatly refused to humor their approach (Matthew 7:6). He has, however, promised them the sign of Jonah. In explaining what He means, Christ gives the first mention of His own death and resurrection as recorded by Matthew. Jonah was the Old Testament prophet told by God to go to the wicked city of Nineveh and preach against it (Jonah 1:1–2). He refused and ran in the opposite direction (Jonah 1:3–4). Jonah ended up being swallowed by a huge fish (Jonah 1:15–17), repenting from his disobedience, and then being deposited by the fish on dry land three days later (Jonah 2:10).

Jesus says a miracle with parallel elements will happen as part of His own ministry. He will spend three days and three nights in the heart of the earth. Jesus doesn't mention the resurrection specifically, but He implies He will no longer be in the earth after three days.

Some have objected that Jesus was not truly buried for three days—as in a full 72 hours of time. However, Jewish people counted every part of a day as one day. In normal conversation, we often do the same today. Most likely, Jesus was buried on Friday before sundown, and left the tomb after sunrise on Sunday morning, the third day. Complaining that this doesn't match His prediction is to miss the miracle involved.