Downward Causation
Given the concept of emergence the universe can be seen as layered. The bottom level is the basic particles of the universe. From that emerge molecules which can be seen as another layer. From that emerges states of matter like solids and liquids. If you keep looking you'll see more and more types of emergence indicating more and more layers to the universe.
Higher layers are composed from lower layers. So if you consider molecules for instance, it's obvious that if you manipulate the atoms in the molecule it will have an effect on the molecule as a whole. Push one atom, If you don't tear the entire molecule apart, then the entire molecule will move. It's pretty easy to see that changes at the lower level will cause changes at a higher level. This is upward causality.
The question for this page is whether downward causality exists. Is it possible for a higher level emergent property to have an effect on and control matter at a lower level?
Reductivist, Non-reductivist, Eliminativist
A reductivist materialist is one who believe that all properties on emergent systems can be reduced to properties on the constituent parts. For example the weight of a rock can be found by adding all the weights of all the molecules in the rock. Weight is not an emergent property, but a reductivist would believe this kind of relationship extends to emergent properties. For example, people have consciousness, and if you look very closely the molecules must have little bits of consciousness in them.
An eliminativist materialist is one who believes that higher level emergent properties don't really exist, we are just confused or fooled into thinking they are there. For example, there just appears to be consciousness but in reality consciousness is an epiphenomenon that has no real effect.
A non-reductivist materialist is then someone who sees emergent properties, but does not believe those properties can be reduced to properties on the constituent parts. Thus air pressure is a property of a gas, but pressure is not found as a property on the individual molecules that compose the gas.