Origin of the word:
Artesian wells were named after the former province of Artois in France, where many artesian wells were drilled by Carthusian monks since 1126. The technique was also known much earlier in Syria and Egypt, although whether the monks of Artois learned of it from outside sources, or discovered it independently,
is unknown.

An Artesian Well is a deep drilled well through which water is forced upward under pressure. The water in an artesian well flows from an aquifer, which is a layer of very porous rock or sediment, usually sandstone, capable of holding and transmitting large quantities of water. The geologic conditions necessary for an artesian well are an inclined aquifer sandwiched between impervious rock layers above and below that trap water in it. Water enters the exposed edge of the aquifer at a high elevation and percolates downward through interconnected pore spaces. The water held in these spaces is under pressure because of the weight of water in the portion of the aquifer above it. If a well is drilled from the land surface through the overlying impervious layer into the aquifer, this pressure will cause the water to rise in the well. In areas where the slope of the aquifer is great enough, pressure will drive the water above ground level in a spectacular, permanent fountain. Artesian springs can occur in similar fashion where faults or cracks in the overlying impervious layer allow water to flow upward. Water from an artesian well or spring is usually cold and free of organic contaminants, making it desirable for drinking.
- ARTESIAN WELL., The Columbia Encyclopedia, Sixth Edition 2006



Why is a well artesian?
In recharging aquifers, this happens because the water table at its recharge zone is at a higher elevation than the head of the well. Pascal's law predicts the theoretical hydrostatic pressure P

    P = rho X g X (z_recharge-z_wellhead})

where rho is the density of the fluid, g the acceleration due to gravity, and z is the elevation. In practical terms the pressure is expressed as metres water column (zrecharge - zwellhead).

"Fossil water" aquifers can also be artesian if they are under sufficient pressure from the surrounding rocks. This is similar to how many newly tapped oil wells are pressurized.
- Wikipedia