What is the difference between slice, segment and leg?
I'm pretty sure those terms are not used consistently across all organisations in the air travel industry. 

They are not defined in an IATA glossary I looked at but I did find this:


  Definition of Flight Segment
  
  The International Air Transport Association (IATA), the international trade body for airlines around the world, defines a flight segment as the operation of a flight with a single flight designator between the point where passengers first board an aircraft and the passengers' final destination. A flight designator includes an airline code, which has two letters or a number and a letter in combination, and a flight number of up to four digits. A flight segment can include any number of stops where passengers board and deplane the same aircraft operated by a single airline.
  
  Comparison of Flights and Legs
  
  A flight is defined by the IATA as the operation of one or more flight legs with the same flight designator. Unlike a flight segment, a flight may involve one or more aircraft. The IATA defines a leg as the operation of an aircraft from one scheduled departure station to its next scheduled arrival station. A flight segment can include one or more legs operated by a single aircraft with the same flight designator.


From eHow