Can you explain to a layman the differences between the former two?
Minkowski spacetime is flat; it has zero curvature. De Sitter spacetime is curved; specifically, it has the same positive scalar curvature at every point.
There is also anti-de Sitter spacetime, which has the same negative scalar curvature at each point. These three spacetimes thus exhaust the possibilities for spacetimes of uniform scalar curvature.
Minkowski spacetime is the spacetime of Special Relativity. Since it has no curvature, it cannot explain gravity as curvature like the curved spacetimes of General Relativity can. Note that de Sitter and anti-de Sitter spacetimes are very special curved spacetimes because their scalar curvature is the same everywhere. In general, the curvature of a spacetime can vary from point to point.
According to the currently accepted cosmological model, our universe is evolving toward a de Sitter geometry as it expands faster and faster. This is because dark energy is becoming the dominant contribution to the large-scale energy density of the universe. Since the density of dark energy stays constant as the universe expands, all other forms of energy are slowly becoming negligible in comparison. As the energy density approaches a constant value, the scalar curvature approaches a constant value.