Traditionally the term "first press" olive oil referred to the oil which was produced from the first crushing of the olives.  It is based on a method of producing oil that is rarely used today when olive paste was placed between mats and crushed with a hydraulic press to produce oil. This was ”first pressed” oil that produced the best quality. The remaining paste was later mixed with hot water and pressed again to produce more olive oil. The oil from this second pressing was inferior, as many nutrients and flavors were removed.  
Today, olive oil is extracted in only one press. 
 
Here in Israel, the term has come to refer to the olive oil in the first month or two since the pressing before it has gone through the natural filtering or "settling" process whereby the fruit particles settle to the bottom.  
 
At this stage the oil is typical not clear, but thick and cloudy and has a more dominant taste and aroma.  First press olive oil is healthier as the oil has more polyphenals than oil which has gone through the filtering process. First press oil is mostly recommended for seasoning and salads.  It is not recommended to be used for high cooking temperatures, until it has "settled".

First Press Oil

Kerem Avital