We know since Hubble's time that the universe expands. Note that this is different than "normal" expansion, which occurs into an already existing medium - it is the space-time itself that is expanding (and that does not prevent it from being infinite! It is like an infinite piece of rubber that is constantly being stretched). However, at the same time, we know that the universe contains matter (such as galaxies), and that this matter gravitationally attracts. As Einstein himself showed, this attraction can lead to collapse of space-time - on a much smaller scale, this leads to the formation of black holes.
Thus, during most of the 20th century, people thought that it is a question of whether the pull of gravity will "win" at the end - in which case the universe will eventually collapse in a "big crunch", or not. Interestingly, there are ways of measuring it - the fate of the universe depends on the average density of matter in the universe, which can be estimated by various methods.
However, in 1999 came a big twist in the story. Two independent groups, led by Saul Perlmutter, Brian Schmidt and Adam Riess discovered that the universe is not only expanding: the expansion of the universe is accelerating, as if some unknown force is pushing away the universe. The origin of the acceleration is unknown; it is named "dark energy", and it contains about 70% of the total energy of the universe.
We know very little about this "dark energy". We do not know its origin, and whether it evolves in time or not. We just know it is there today. If it is fixed in time, the space will continue to expand at faster and faster speeds, and eventually nearly all the galaxies will disappear from our sight. The only exception is our closest neighbor- the andromeda galaxy, which is so close to our own milky way that it is now falling towards it due to its gravitational force. The two galaxies will eventually collide and merge, but this will happen only in about 4 billion years from now.
Thus, if indeed dark energy is fixed with time, then indeed the universe will dissolve into nothing; but there is plenty of time before this happens...