Greetings de-emerald,
It is always refreshing when someone wants to truly know about these events and is my good pleasure to explain them, as well as all other Biblical topics. And though it is my policy to include scripture, I will try to refrain per your request:
Q: Explain the Rapture and the tribulation
The Dead in Christ - The word Rapture "caught up," is where Paul stated that the Lord would descend to the atmosphere (the air) and would resurrect the bodies of those who will have died from beginning of the church until that event takes place, where at which time their spirits will be reunited with their immortal and glorified resurrected bodies and caught up in the air.
The Living in Christ - Immediately after the dead in Christ have been raise in the air, those who are still alive in Christ, will be transformed into their immortal and glorified bodies and will be caught up with those who will have just been resurrected. I believe this entire process of the resurrection and the living being caught up will take place in an instant. At this point the entire church from beginning to end will be gathered in the air to meet the Lord where according to His promise, will take the entire group back to the Father's house to those places that He went to prepare of us.
Tribulation - The tribulation period, is another name for 'The Day of the Lord' which is the time period of God's wrath and will be seven years length and which follows A & B. The book of Revelation is a detailed account of that time of tribulation which will be carried out via the seals, trumpets and bowl judgments, which are three sets of seven judgments.
Does the tribulation happen immediately after the rapture - Once the church, dead and living, have been caught up to meet the Lord in the air and are taken to heaven, those who are still on the earth will be locked in to that time period which will begin when that ruler, the antichrist establishes his seven year covenant with Israel, which will allow them to build their long awaited temple and begin to temple sacrifices according to the Law. How long in between when the church is gathered and the seven year covenant is made, scripture does not say, but I'm pretty sure that there will not be that much of a gap. The world is going to be in chaos because of all those who are missing and this would be the perfect time for that antichrist to reveal himself. It is during this time that the seals will begin to take place, followed by the trumpet and then the bowl judgments, as well as the plagues of wrath that the two witnesses bring, which again will last through the entire seven years.
What happens to non-believers in the Tribulation - Non-Believers, the unsaved, will be those who will end up worshiping the beast, his image and receiving his mark in order save their physical lives. I'm sure that many of them will be killed via the seals, trumpets and bowl judgments through the entire seven years. When the Lord returns to the earth to end the age at the end of the seven years, those wicked who make it alive when the Lord returns will be killed by the Lord and their bodies will be eaten by the birds that the angel will have gathered for that very purpose. Anyone who worships the beast, his image or receives his mark, will be tormented in fire and sulfur (Lake of fire) in the presence of the holy angels and of the Lamb. And the smoke of their torment will rise up forever and ever. And they will have no rest Day and night.
How long the tribulation is for - Sorry but I have to mention a scripture LOL ... : In Daniel 9:27 seven years is the last seven years of seventy seven year periods (490) that was decreed upon Israel and Jerusalem. Sixty nine of those seven year periods have been fulfilled. One seven year period remains unfulfilled, which will be fulfilled in conjunction with the Lord's physical and visual return to the earth to end the age and establish His millennial kingdom. It is shortly after the 7th bowl judgment has been poured out, which completes God's wrath, that Jesus will return to the earth. So to recap, the tribulation period is seven years in length, with the last 3 1/2 years referred to as 'The Great Tribulation.'
What happens after the Tribulation - After the tribulation, the Lord will return to the earth to end the age and establish His millennial kingdom on this earth. He will have the beast and the false prophet captured and thrown alive into the lake of fire. The wicked will be killed by the double-edge sword which proceeds form the Lord's mouth, which figuratively represents the spoken word of God. An angel having the key to the Abyss will seize Satan and will cast him in and seal it over him where he will be restricted during the thousand year reign of Christ. At that time, all of those tribulation saints who will have died, will be resurrected immortal and glorified, being part of the first resurrection, and they will rule with Christ during the thousand years. There will be pre-millennial business such as the judgment of the sheep and the goats. The Lord will settle disputes between the nations of the righteous who make it through the tribulation period and who along with the remnant of Israel, still being in their immortal bodies, will repopulate the earth during that thousand years. The length of age will be extended during that thousand years like it was prior to the flood. I'm sure that there will be other pre-millennial business that will take place which I haven't mentioned here.
What Happens After the Thousand Years - I know that you didn't ask this one, but I'm adding it. After the thousand years if over, Satan will be released from the Abyss and will go out to deceive many who will have repopulated on the earth, figuratively called Gog and Magog and they will surround the city God loves, Jerusalem and God will rain fire down upon them.
Then Satan will be taken and thrown into the Lake of fire where the beast and the false prophet will have been thrown a thousand years earlier, where they will be tormented day and night forever and ever.
After this, the spirits of the wicked throughout all of history will be released from Sheol/Hades and will be given resurrected bodies and will stand before God and be judged before His great white throne to give account of all of their sins. In addition, it is at the great white throne judgment where death and Hades personified, will be thrown into the lake of fire. Anyone's name not found written in the book of live will be cast into the Lake of fire.
Sometime after the Great White Throne Judgment, God will will destroy this current heaven and earth and all of their elements and will create a new heaven and new earth, with the new Jerusalem coming down from God out of heaven. This is the eternal state where those whose names were found in the book of life will live in the new Jerusalem and the new earth.
I hope that this was basic enough for you