An atom contains a dense nucleus orbited by negatively charged electrons. The nucleus of the atom contains two types of particles, protons and neutrons (hydrogen atoms have a single proton). Protons have positive charge and neutrons do not carry any charge. The way that atoms combine to make molecules and then the way that molecules combine to make materials, can be quite complicated. However, in some materials this results in the presence of free electrons.
A free electron is an electron that is not attached to its parent atom and is free to wander around the material. A free electron is a mobile charge carrier. The number of free electrons in a material determines its resistivity and therefore is the main factor in determining the material's resistance. It is these free electrons that move, when a voltage is applied, to produce current flow. If there are no free electrons, then there cannot be any current flow. If there are only a few free electrons, the current will be very small and we say the resistance of the material is high. If there are a lot of free electrons, the current can be quite large, we say that the material has low resistance (or high conductivity).
Number density is a measure of the number of free electrons per cubic metre of material. The number density in conductors is very high, in the order of 1028 free electrons per cubic metre.