An ohmic contact is a highly-conductive junction between a metal and a semiconductor. Ohmic contacts are used by the millions in integrated circuits to contact the gates, sources and drains of MOSFET's. Although it sounds trivial, forming low-resistance ohmic contacts was a difficult problem for the early semiconductor researchers. The problem is that unless the silicon is heavily-doped, the contact is almost sure to be a Schottky diode. If you look at the diagram of a MOSFET, you will see that the source and drain are heavily-doped "n+" or "p+" regions.
Although in theory it seems that an ohmic contact to an n-type semiconductor can be made by simply using a metal with work function larger than that of the semiconductor*, it turns out that charge traps at the interface cause such junctions to be Schottky diodes. In practice, the only way to make an Ohmic contact is to use a heavily-doped semiconductor. The ohmic contact works by a quantum mechanical process called tunneling. The depletion region in a metal/heavily-doped-semiconductor junction is very thin . Because of their wave nature, electrons can tunnel through the depletion region, allowing current to flow. The doping needs to be higher than 1019/cm3 (1020/cm3 is much better) to form a decent ohmic contact.
* or opposite for a p-type semiconductor
Ohmic contacts can have non-neglible resistance. In a process I use, the contact resistance of a 2μm x 2μm square aluminum/silicon junction is 50Ω. This contact resistance seems fairly small but it results in a 5-10% drop in current. As integrated circuits shrink, contact resistance increases due to smaller contact area. While it's tempting to overlook ohmic contacts, they have real-life degrading effects on circuit performance.