This article is considered accurate for the current version of the game.
Achievements are unlocked for players after reaching certain milestones, achieving hidden goals, or unlocking certain items. There are currently 62 unlockable achievements in Cities: Skylines.
Active mods (not assets) will disqualify a save game for achievement completion. Accepting a cash bail out will disqualify a save game for achievement completion. Although there is a Steam Workshop modification that makes it possible to enable achievements.
See Global Gameplay Stats – Cities: Skylines at Steam for a list of all achievements sorted by the percentage of players that managed to complete an achievement.