The expanding universe theory states that the physical matter and energy of the universe has been expanding continually since the Big Bang. This means that the size of outer space is potentially infinite, even as the matter and energy inside that space is finite. The fate of the universe depends on whether the expanding model is open, closed, or flat. For this question, an open expanding model is assumed.
First, since matter and energy are finite and constant, the separation of galaxies and stars will cause the attraction of matter by gravity to lessen. As all the matter that affects other matter moves outward, the ability of a galactic core to keep its stars centered will become weaker. As this occurs, the average temperature of the universe will even out. An average temperature for everything in the universe will mean that stars and warm planetary cores are impossible. This is known as the "Heat Death" of the universe, where every particle is the same temperature and so no reactions can take place.
Second, if there is accelerated expansion based on various theories of "Dark Matter," which relies on a type of energy which cannot be understood, all matter and energy will continue to separated from itself forever. This would mean that every particle, whether matter or energy, is separated from each other particle by a space so large that it is essentially infinite. As in the first scenario, since there can be no collection of matter by gravity because the distances are too large, the structure of the universe can be said to be "ripped apart," giving this fate the name of "The Big Rip."