In condensed matter physics, often we are interested in the flow of charge from a "source" electrode, through some material (the "channel", probably a different material than the source), and away into a "drain" electrode. When the source, channel, and drain are all metals, life is simple. While there might be some mismatch in the electrical conductivities of the materials, in the end an electron can go from some delocalized (extended, wavelike) state in the source, into such a state in the channel, and then into such a state in the drain, smoothly. Because of the discontinuity in electronic band structure and dielectric properties, there is some reflection at the interface, leading to a contact resistance. That is, some fraction of the applied voltage, linearly proportional to the applied voltage, is dropped across the source-channel contact, and some across the channel-drain contract. This is an example of an ohmic contact.
However, the situation can be more complicated. If the channel is a crystalline semiconductor, the Fermi level of the metal usually winds up sitting somewhere in the band gap. If the is appropriate band bending takes place, there can then be an energy barrier (a Schottky barrier) for injection if charge from the metal into the semiconductor. The spatial width of the barrier depends on the level of doping in the semiconductor, with higher doping leading to a narrower (though not necessarily shorter) barrier. In this case, the current-voltage characteristics of the contact is not Ohmic, and looks instead like a diode, because the applied bias changes the shape of the barrier. To avoid this in transistors, the regions of the channel where the source and drain contact it are very highly doped. Still, in this case we are still assuming that the actual electronic states are extended, delocalized things.
The situation gets more complicated when the channel does not have delocalized states near the Fermi level.
Usually experiments are designed to mitigate contact effects, either by avoiding measurements of the contact voltages (so-called four terminal measurements) or by making the contact contribution negligible compared to the bulk channel. However, it turns out that sometimes contact effects can provide valuable insights into charge transport properties in the bulk. I'll write more soon about this.