I spent several hours searching the Internet for an explanation of "the last day" that made sense and that fit all of the pertinent verses. Unfortunately, I found very few explanations of the meaning of this term, and none of the explanations that I found met these criteria, including the most popular explanation in this forum, which claims that "the last day" refers to "a specific 24-hour day" when the rapture occurs. I have to disagree with this explanation for three reasons.
One reason is that the Greek word that John used 30 times for "day" is the exact same word that Paul used in 1 Thess. 5:2 for "day" in the phrase "the day of the Lord" and the exact same word that Peter used in 2 Pet. 3:10 for "day" in the phrase "the day of the Lord." Both Paul and Peter stated that "the day of the Lord will come like a thief in the night," which is a direct reference to the rapture because Jesus used an almost identical phrase to describe the unexpectedness of His coming in Matt. 24:43. However, in the same verse (2 Pet. 3:10), Peter also mentioned the destruction of heaven and earth that will occur on this same day. As we know from Rev. 20:1-21:1 this won't happen until the end of the Millennium. This means that "the day of the Lord" will last approximately 1007 years (not 24 hours) because it begins at the rapture and ends after the Great White Throne judgment when our present heaven and earth are replaced with a new heaven and earth.
The second reason I disagree with the author's claim is that the resurrection of the "dead in Christ" (1 Thes. 4:16) that will immediately precede the rapture is a different resurrection event from the resurrection of the saints who will be martyred during the "Tribulation Period." (John mentioned this second resurrection event in Rev. 20:4-5 and refers to it as "the first resurrection," which post-tribbers incorrectly interpret as the same resurrection event that will occur at the rapture because of the word "first.") Surely, Jesus also had these saints in mind when He said "I will raise them up at the last day."
The third reason I disagree with the author's claim is that the judgment of those who reject Christ will occur "at the last day" as well. This is clear from the statement Jesus made in John 12:48:
“He who rejects Me, and does not receive My sayings, has one who judges him; the word I spoke is what will judge him at the last day." (NASB)
John described this judgment in Rev. 20:11-15, which will not occur until after Jesus has been reigning on the earth for 1000 years. While it's possible to interpret "the last day" in John 12:48 as the last 24-hour day of human history before eternity begins, it makes more sense to interpret it as the same "day" on which the rapture and the "first resurrection" occur, which is "the day of the Lord" that will encompass all of these events, according to 2 Pet. 3:10.
To summarize, "the last day" refers to an extended period of time (aka, the "day" of the Lord) that includes the rapture, the tribulation, the Second Coming, the Millennium, and the Great White Throne judgment." This coming "day" is also the last or seventh "day" of human history, which will be a total of 7000 years per the statement "one day is as a thousand years, and a thousand years is as one day" in 2 Pet. 3:8. We are currently approaching the end of the sixth "day," which is equivalent to the end of 6000 years.