The millennium is the period of time that Jesus reigns as King. There is debate as to the nature of the millennium. Is it a literal 1000 years or is it a figurative length of time? Below is a chart that simply lays out the two dominant positions: premillennialism and amillennialism.
Premillennialism is the teaching concerning the end times (eschatology). It says that there is a future millennium (1000 years as mentioned in Revelation 20) where Christ will rule and reign over the earth. At the beginning of the millennium Satan and his angels will be bound and peace will exist on the entire earth. At the end of the 1000 years Satan will be released in order to raise an army against Jesus. Jesus will destroy them and then the final judgment will take place with the new heavens and the new earth being made.
Amillennialism is the teaching that there is no literal 1000 year reign of Christ as referenced in Revelation 20. It sees the 1000 year period spoken of in Revelation 20 as figurative. Instead, it teaches that we are in the millennium now, and that at the return of Christ (1 Thess. 4:16 - 5:2) there will be the final judgment and the heavens and the earth will then be destroyed and remade (2 Pet. 3:10).