It's not as naive a question as you may think, and the answer is a lot more complicated than you may think.
When we're calculating how the universe expands we assume it's isotropic and homogeneous (this just means on average it's the same everywhere) and it has a scale factor that is normally written as $a(t)$. The scale factor tells us how much the universe has expanded. We set the scale factor to be 1 at the current time, so a scale factor of 2 means everything is twice as far away and a scale factor of 0.5 means everything is half as far away.
If our scale factor $a(t)$ was constant then the universe would be static i.e. distant galaxies would be stationary with respect to us. When we say the universe is expanding we mean that the scale factor $a(t)$ increases with time. To find out how $a(t)$ changes with time we have to solve Einstein's equations, and this is where things get messy because the solutions are complicated. If you're interested have a look at the Wipedia articles on the FLRW metric and the Friedmann equations. Without going into the details, we expect the scale factor to look something like:
Without dark energy the rate of increase of the scale factor gradually decreases with time because the mutual gravitational attraction of all the matter in the universe slows the expansion. With dark energy the expansion rate is always slightly higher, and at large times the expansion rate starts increasing again.
The original detection of dark energy was based on measuring the recession velocity of supernovae, and discovering that they matched the predictions from the red line not the black one.
A few additional notes: we believe the universe is flat, and this means that (in the absence of dark energy) the expansion rate shown by the black line would continue to slow but would never actually become zero. More precisely it would tend asymptotically to zero as time tends to infinity. An open universe means the expansion rate would tend to a value greater than zero, and a closed universe means the expansion rate would reach zero in finite time then become negative. A closed universe would start contracting again.
Also note that at time zero the scale factor is zero i.e. the distance between everything in the universe would be zero. This point is what we call the Big Bang. The name is misleading because it wasn't an explosion (as so often shown on popular science TV programmes). It's actually a singularity because if the spacing between everything is zero the density must be infinite. A singular point is where our equations break down because we can't do arithmetic with the number $\infty$.