Use the noun advancement when you talk about something's growth or progress. The advancement of modern medicine has lengthened many people's lives.
You might describe a society's advancement as a result of new industries and an improved economy. You could also cheer your best friend's advancement in her French class, being moved to a higher level because of her hard work. When you advance, you move forward — either literally or figuratively. Advancement comes from this idea of moving in a positive direction and its Old French root word avancier, "move forward," from the Late Latin abante, "in front."