What are the criteria for classifying which generation a fighter aircraft belongs to?
The Wikipedia Page for Fighter Generations has a relatively comprehensive list on how they derive at the "generation" of the fighter. For example, here is how the Air Force classifies them:


  
  1: Jet propulsion : F-80, Me 262
  2: Swept wings; range radar; infrared missiles : F-86, MiG-15
  3: Supersonic, pulse radar, Beyond-visual-range missiles : Century Series, F-105, F-4, MiG-17, MiG-21
  4: Pulse-doppler radar; high maneuverability; look-down/shoot-down missiles : F-15, F-16, Mirage 2000, MiG-29.
  4+: High agility; sensor fusion; reduced radar signature : Eurofighter Typhoon, Su-30, F/A-18E/F, Rafale
  4++: Active electronically scanned arrays; continued reduced signatures or “active” (waveform canceling) stealth technology, supercruise : Su-35, F-15SE
  5: All-aspect stealth with internal weapons bays, extreme agility, full-sensor fusion, integrated avionics, some or full supercruise : F-22, F-35,
  6: extreme stealth, morphing capability, smart skins, highly networked, extremely sensitive sensors, optionally manned, directed energy weapons.
  


Until we have a major leap in technology, it would seem we are capped off at 5th generation fighters for the moment.