We aren't quite sure
The problem with determining the fate of the universe, is that it depends on the model of the universe: it size, shape, mass, contents, etc. Most of which we don't fully know.
However, there are three predicted models so far:
Big crunch: the acceleration causing the universe to expand will begin decrease and gravity starts to take over and time seems to 'undo' itself as the universe comes together to a single point.
Big freeze: Possibly the longest of the predicted models. This model says that the expansion of the universe will continue at a fairly constant rate or reach a maximum size, and the universe carries on as normal. Though stars will slowly die out, and the resources necessary for new stars will be become more scarce, and less and less stars will form. All current stars will die off in a way respective of their type, i.e. nebulae, black holes, white dwarfs, etc.
From there, the heat death phase will begin where the entropy of the universe will reach a maximum, or thermodynamic equilibrium will be reached, and so no processes can happen on their own.
At that point we won't know what will happen because as far as we know, protons don't decay, and at the quantum level anything could still happen (like virtual particles, quantum fluctuations, anything related to string theory, etc.).
A more ordered timeline can be found here
Big rip: This model says that dark energy will continue grow stronger with time and cause the expansion to accelerate. Such an event will cause large bodies like galaxies to recede from each other quickly, though soon the very galaxies will tear themselves apart, and solar systems, star systems, individual objects (stars, planets, moons, etc.), and the sub-atomic particles in the nuclei themselves. The expansion will be so great that objects will be moving faster away from each other than light itself. So, the universe will be pretty filled with individual particles that will mostly decay, for example neutrons, and separated by infinite distances.
According to the latest empirical evidence the universe will end in a state of maximum entropy with nothing left
The 1997 and 1998 studies of supernovas provided empirical evidence that the rate of expansion of the universe is increasing.
This means that omega is greater than one and there is not enough gravity or matter to reverse the flow of entropy and the universe will continue to expand until there is no useable matter or energy left in the universe.
This means that our present observable universe will have an ending just as the empirical evidence indicates that the universe had a beginning. The philosophical implications are that our universe is not eternal, nor self existent, and something besides matter and energy must exist.
Look up the Supernova Cosmology Project and the High Z Supernova Search Team. Members of these research teams received a Noble prize for their discovery.