If I'm not mistaken, a blank is a round that has no projectile but still a significant amount of gun powder in it, which makes the loud noise when it's ignited by the primer that sounds similar to a real round going off. That gun powder is still under pressure, so it can still fire out of it any projectile that might be in the gun, like debris. I've heard of a case of someone putting a gun filled with blanks to their head and firing as a joke thinking it was harmless, but the pressure of the gunpowder had nowhere to go with the barrel pressed firmly against their skin so the person's skull actually fractured and essentially became a projectile itself, killing them.
In the case of Brandon Lee, they had earlier removed the gunpowder from cartridges to create a "dummy round" which has no gun powder but still has a primer and a projectile, so it looks authentic. The trigger was pulled with the dummy rounds in at some point but with no gun powder the bullet didn't fire completely but apparently there was still enough pressure that the bullet left the cartridge and became lodged in the chamber and nobody noticed. The gun was then loaded with blanks and when they were fired during the scene, that round caught in the chamber was fired, killing Brandon Lee.
Not sure what happened in this case here but I highly doubt it's anything intentional, just negligence/bad accident. I doubt Alec Baldwin is charged with a crime. He is the world's greatest actor, after all.