An artesian aquifer is a confined aquifer containing groundwater under positive pressure. This causes the water level in a well to rise to a point where hydrostatic equilibrium has been reached. This type of well is called an artesian well. Water may even reach the ground surface if the natural pressure is high enough, in which case the well is called a flowing artesian well.
An aquifer is a geologic layer of porous and permeable material such as sand and gravel, limestone, or sandstone, through which water flows and is stored. An artesian aquifer is confined between impermeable rocks or clay which causes this positive pressure. The recharging of aquifers happens when the water table at its recharge zone is at a higher elevation than the head of the well.
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.