An Eternal UniverseIf the Universe is Open or Flat then it will continue to expand for eternity. Or put more simply, if the universe does not contain enough matter to stop its expansion it will continue to expand forever. The difference between the Open and Flat Universe is subtle. In the Flat Universe there is exactly enough matter to bring the expansion to an end when time has reached infinity. In the Open Universe there is not enough mass to stop the expansion ever. Since time equal to infinity is forever it is correct to say that in both types of Universe the expansion shall continue for all eternity.
Using the currently understood laws of physics we can project into the future what the Universe may look like in very distant eras. Two astrophysicists at the University of Michigan, Fred Adams and Greg Laughlin, have written a paper outlining the future history of the Universe. For convenience they invented a unit of time called the Cosmological Decade. A cosmological decade of 20 is 10^20 years (or 1 with 20 zeros after it).
They have further divided the future into Eras. The current Era is known as the Stelliferous or Star-Filled era. In this era the Universe is filled with stars and galaxies and planets as it is today. This era lasts from Cosmological decade 6 to 14 (that's 10^14 years or 100 Trillion years). At the end of this era all stars have exhausted their fuel and have died leaving behind only remnants of their once glorious era.
The next era is known as the Degenerate era. In this era the universe is made of dead planets, brown dwarfs, white dwarfs, nuetron stars, black holes, and some theoretical forms of dark matter. These objects still gravitationally interact with one another occasionally colliding and forming new stars. This era lasts until cosmological decade 37. At the end of this era all protons, which compose the nuclei of all atoms, degenerate.
Cosmological decade 38 - 100 is known as the Black Hole era because black holes will be the only gravitaionally important objects left in the universe. However, Black holes do not last forever. They evaporate under what is known as Hawking radiation. By cosmological decade 100 all black holes will have evaporated.
From cosmological decade 100 on the Universe shall be composed of only electromagnetic radiation and particles which have an infinite lifetime, so far as anyone can tell, such as electrons, positrons, and neutrinos. From this point on interesting things might continue to happen but we have essentially reached the limits of our physical knowledge.