(Please note: this is a brief history. There might be additions in the future if they become pertinent. Or if we think of something really, really cool.)
The Avengers were a group of heroes that banded together in 2002. It happened organically as individuals sometimes banded together to take on multiple super villains or actual groups of them. They decided a group of powered heroes was needed to help keep people safe. After a couple years, they gained official recognition and sanction from the American government and not long after, sanction from the UN to operate internationally.
In 2009, after Registration officially becomes a US law, the Avengers disband.
In 2015, SHIELD starts to unofficially gather together heroes to combat evils in various parts of the world. Sentinels are a large part of that but not the only one.
In 2017, after a shift in power in the US political system, Registration of powered citizens is repealed. The Avengers reform, its membership made up of many of the heroes who were working with SHIELD and one or two of the original members. They are not affiliated with SHIELD nor the American government, prefering to retain their autonomy.