Bronze Trophies:

Franchise – Win the NBA Finals
Win the NBA title – any year, any skill setting
   
Quick Play – Defeat 5 NBA Teams – All-Star
Defeat 5 NBA Teams – min All-Star difficulty setting, 5 min + qtrs – no spotting points
   
Mini Games – Rabbit
Defeat opponents (Overall rating 85+) in a single game – Minimum All-Star difficulty setting
  
Mini Games – Blacktop Golf
Score 70 or less points – Minimum All-Star difficulty setting
   
Mini Games – 1-on-1
Defeat opponent (Overall rating 85+) by 5 or more points – Minimum All-Star difficulty setting
   
Mini Games – 2-on-2
Defeat opponents (Overall rating 85+) by 5 or more points – Minimum All-Star difficulty setting
   
Mini Games – 3-on-3
Defeat opponents (Overall rating 85+) by 5 or more points – Minimum All-Star difficulty setting
   
Mini Games – Own The Court
Defeat opponents (Overall rating 85+) by 8 or more points – Minimum All-Star difficulty setting
   
Mini Games – 3-Point Shootout
Defeat opponents (Overall rating 85+) by 5 or more points – Minimum All-Star difficulty setting
   
Mini Games – PlayStation Skills Challenge
Defeat opponents (Overall rating 85+) by 3 or more seconds – Minimum All-Star difficulty setting
   
NBA Replay: The Season – Complete First Half of Mode
Complete First Half of Mode – All Star difficulty setting
   
Player Progression – Advance 5 to 70+
Advance 5 Progression Players to an Overall Rating of 70+ points for each player

Online 5 Game Winner
Win 5 online games

Silver Trophies:
  
The Life – Story #1 Completed
Complete Story #1 – Minimum Veteran difficulty, Complete all “Extra Mile” goals
   
The Life – Story #2 Completed
Complete Story #2 – Minimum Veteran difficulty, Complete all “Extra Mile” goals
   
The Life – Story #3 Completed
Complete Story #3 – Minimum Veteran difficulty, Complete all “Extra Mile” goals
 
Franchise – Win 3 NBA Finals
Win 3 NBA titles – any years, any difficulty setting
 
Quick Play – Defeat 15 NBA Teams – All-Star
Defeat 15 NBA Teams – min All-Star difficulty setting, 5 min + qtrs – no spotting points
   
NBA Replay: The Season – Complete Entire Mode
Complete Entire Mode – All Star difficulty setting
   
Player Progression – Advance 5 to 80+
Advance 5 Progression Players to an Overall Rating of 80+ points for each player
   
Online 10 Game Winner
Win 10 online games
   
Gold Trophies:

The Life – All 3 Stories Completed
Complete all 3 stories – Minimum Veteran difficulty, Complete all “Extra Mile” goals
   
Franchise – Complete 20 year Franchise with 5+ NBA Titles
Complete 20 year Franchise with 5+ NBA Titles – any difficulty setting
   
Quick Play – Defeat all 32 NBA Teams – All-Star
Defeat 30 NBA Teams + All-Star Teams – min All-Star difficulty setting, 5 min + qtrs – no spotting points
   
NBA Replay: The Season – Complete Entire Mode including all Extra Mile Goals
Complete Entire Mode including all Extra Mle Goals – All Star difficulty setting
           
Player Progression – Advance 5 to 90+
Advance 5 Progression Players to an Overall Rating of 90+ points for each player
           
Online 25 Game Winner
Win 25 online games
           
Platinum Trophy:

100% Clear
This is unlocked automatically when all the trophies have been unlocked

Source: ps3trophies.co.uk