The achievements provide an additional series of challenges to complete in order to gain many rewards.
Achievements are separated in tiers, each with their own rewards. Once you complete a tier, another one replaces it until you complete all tiers of the achievement.
Rewards include MONOPOLY money, diamonds and tokens.
Once a reward can be claimed, a notification appears on the Tycoon HQ button and on the Achievement button.
You can unlock the achievement's system around city level 4 in Atlantic City.