Eschatology is the study of end times. This is not limited to Christianity or even religions as a whole. It can also refer to study of the ending of the physical universe. Some have also applied it to the study of a time when technology plateaus at an almost magical level, sometimes referred to as "the Singularity" (sometimes referred to as "the Eschaton").
Because eschatology deals with the future, it is difficult to have much certainty. Cosmologists are certain the universe will run down, and Christians are certain Jesus will return. Additionally, the Bible passages which teach on eschatology use a considerable amount of symbolism.
As a result, faithful Christians can, and do, disagree on the particulars of eschatology.
Normally, these issues are minor, and can be overlooked. But sometimes, it does matter.
The greatest difference between orthodox Christian eschatologies would be postmillennialism (post-mil) and premillennialism (pre-mil). Post-mils believe Jesus will return after all the earth has come to faith in Jesus (effectively, achieving "heaven on earth"). Pre-mils believe that Jesus will come suddenly.
The "millennial" refers to the one thousand year reign of Christ (Revelation 20:4).
Some Christians believe this will be a literal thousand year reign. Either on the current earth, or some pre-eternal new earth, or even in heaven (before returning to create the new earth).
After this is "the eternal state". Amillennialists do not believe in a literal thousand year reign. If they are agree with post-mils, then the thousand years refers to the time before Jesus comes (He is reigning "through the Church", then Jesus comes). If they are of a more pre-mil bent, then the thousand years is a metaphor for eternity.
Historically, amillennialism and post-millenialism have been most popular. I lean toward pre-millenial (dispensationalism), which is relatively new (being only one hundred years old).