A verified Instagram account (marked with blue badge/check mark) means that the selected profile has been verified by Instagram and is confirmed to be an official and authentic account of a celebrity, public figure or a brand.
The ‘verified badge’ shows on the selected profile and in the search feed (it is shown neither in the newsfeed nor on Instagram stories).
Why has Instagram introduced verified accounts?
Simply to avoid having fake accounts of famous people or brands.
Since August 2018 it is actually possible to request verification, and the process itself is very simple.
It may happen that Instagram will take away the blue badge from a selected account (even if it was already verified) if it happens to violate Instagram’s rules and regulations or is no longer a public account.
Another thing worth pointing out is that having a verified Facebook profile doesn’t automatically means having verified Instagram account (although Facebook owns Instagram). It’s actually much harder to get the verified badge on Instagram, which means verification process and guidelines differ a lot when it comes to Facebook and Instagram (e.g. the number of followers is not the most important thing in the verification process on Instagram).