As you note, the different "millennialisms" are based on different interpretations of the "thousand year reign" mentioned towards the end of Revelation. Unfortunately, answering this question is not as easy as simply stating the different views on the millennium itself. Each of the different views only make sense inside their respective theological frameworks. For example, you cannot transplant the amillennial view of the thousand years and into a dispensational theological framework. Thus, in order to properly answer this, we must include bits of subjects other than just the millennium itself and one must be aware that you cannot simply choose one of these views on their own, you must arrive at one of these understandings based on your hermeneutic for understanding the rest of Scripture.
As you noted, there are several main views on eschatology. Your question mentioned three. While no set of boxes will suffice to accurately categorize these views, I think it's unfair to not include a fourth, namely historic-premillennialism. The modern view called premillennialism is radically different from what was known as premillennialism just a few hundred years ago. The radical shift of views under the same name makes it confusing, so one must delineate.*
How each of these views addresses the what and when of the millennium is also tied up in their views of the church, what the kingdom of God is and the timelines of both world history and individual lives.
Postmillennialism
Postmillennialism identifies the millennium as a future event in this world where Christians triumph over this world and the Kingdom of God is visible (as opposed to the church-invisible). Of all the views I understand this one the least, but I believe there is some variance of opinion on when the millennium starts. Some say it will be a gradual event where the church sets right all the wrongs of the world step by step until eventually we reach a golden age. Another view is that Christ will dramatically intervene at some point in the future to bring this about.
The thousand years of the millennium are generally considered to be figurative -- meaning a long period of time but not necessarily exactly 1000 years as we know them. Satan is bound during the millennium both spiritually and physically so he can't wreak his normal havoc. After the thousand years, Jesus will return. Those believers living will be caught up to be with Him in heaven and the dead will be raised, either to life with him or eternal death.
Amillennialism
Like post-millennialism, amillennialism believes that the thousand years is a figurative number indicating a long period of time and that Jesus will return after the thousand years is over. Unlike the other views, amillennialism believes that the thousand years has already started. Beginning with the first coming of Jesus and the establishment of the NT church the reign of Christ in victory through the invisible church has already begun. Satan is currently bound in that his power to stop the spread of the Gospel is limited. The kingdom is growing and reigning victoriously through true believers in the world today. However the world as a whole is also still inhabited by evil and good and evil grow side by side and in contrast to each other until Jesus returns for his own. This is the period of tribulation, and the church experiences but is preserved through it. Jesus second returning ends the millennium and marks the second judgement.
Premillennialism
The modern iteration of premillennialism is based on dispensational theology and is sometimes knows as Dispensational Premillennialism, although if someone gives no qualification as to which premillennialism is being referenced, it is is generally assumed to be this one.
The dispensational premillennial timeline is the hardest to describe. The millennium is a future event in which the nation of Israel is restored and all the promises about her are fulfilled. Before the millennium starts however, Jesus returns two more times, once privately to rapture the gentile believers out of the world and again publicly at the start of the millennium. In between these two returns is the tribulation, a 7 year period of disaster and judgement on earth. There are three views on when the church is raptured in relation to the tribulation, pre-tribulation, mid-tribulation and post-tribulation. Evil in the world will generally increase until the start of the tribulation, at which point all hell breaks loose in a time of satanic dominance. After that time is over and Jesus has come (the third time if you are counting from time he came for the cross) Satan's power will be entirely bound and a new kingdom will be established on earth where the Jewish people are restored to their place as God's chosen people and reign on earth. When that's over eventually the church already raptured out of the world and the restored faithful remnant of Israel will be reconciled in heaven.
Historic Premillennialism
Also known as "Classic Premillennialism", is actually more similar to Amillenialism than it's modern name-sake. The period of tribulation for the church began with its establishment after the resurrection and will end when Christ returns to earth a second time. Evil will increase in the world along side of good. The church invisible will be sorely tried but always be victorious as protected and strengthened by Christ along the way. There are different views on the great tribulation and whether all of the prophecies are spread out throughout this church age or whether some of them are concentrated into a special great-tribulation period during the last seven (literal or figurative) years at the end. Either way, Christ returns once when the tribulation period is over both to collect his church out of the world, resurrect the dead, and to judge evil. Where historic pre-millennialism differs from amillennialism is what happens after that. Amillennialism sees the thousand years as overlaid on top of the tribulation, parallel tracks as it were or different perspectives on the same event. Historic pre-millennialism believes that the tribulation is now but that after the second return there will be a time of earthly reign in which all things are made new, Christs power to restore is demonstrated on the same turf on which the rest of history was played out but with him physically present and Satan bound. This time can be interpreted as either figurative for a long period of time or a literal thousand years.
*For full disclosure in case my bias comes through somewhere: I am historic-premil.
