The force causing galaxies to accelerate away from each other? The galaxies in the universe appear to be accelerating away from each other according to the Doppler shift that is present in the light we receive from other galaxies.
According to Newtonian mechanics for something to accelerate, a force must be applied to it, and if the accleration is constant, then the force must be constantly applied.
If this is the case, then what force or forces is (or are) causing galaxies to accelerate?
 A: Newtonian mechanics does not explain the dynamics of the universe.
The expansion of space is apparently "caused" by an innate property of the vacuum.
The name we give to this property is dark energy. 
In the absence of any gravitating matter, the dark energy content of the universe causes space to expand as a natural consequence of solving the dynamics of the whole universe according to the theory of General Relativity.
In GR, mass, energy and pressure all act to gravitationally decelerate any expansion. Dark energy acts like a negative pressure that causes accelerating expansion, and there is no evidence at the moment that its energy density (i.e. energy per unit volume) changes with time - so it is sometimes referred to as the cosmological constant.
It is still a matter of theoretical speculation at present as to what dark energy is.
However, before observations of the cosmic microwave background and distant type Ia supernovae revealed the need for dark energy, it was still known that the universe expanded and the equations of GR still permit an expanding (although decelerating) universe without dark energy. One idea is that the universal expansion was kicked off by cosmic inflation - a souped-up version of dark energy that caused the universe to expand exponentially for a short period of time in the first tiny fraction of a second after the big bang. 
A: Dark Energy is thought to be responsible for an ever expanding Universe. And we cannot understand this phenomenon based on Newtonian Mechanics as it is not applicable in cosmological scales. You can think of this energy as a negative pressure due to vacuum. But we truly don't know anything else about its nature and characteristics. However, DES survey is under way to understand more of it in the coming decade. 
