"I asked someone in the Bush/Cheney office what this BMG gun was and they told me that it was an anti-air gun to take down planes so I came to FR to ask exactly what it was"
The .50 BMG (gun) is the Browning MachineGun.
The .50 BMG is the also the name of the cartridge for that gun, so it can be confusing.
You as a private citizen cannot order a new .50 BMG for purchase. You can buy legally a single shot or semi-automatic rifle chambered for that cartridge.
A lucky hit that took out the engine of a large aircraft would not bring it down. A fully automatic machinegun is needed to really work in an anti-aircraft role. But, you cannot buy one of those new as an individual anyway. Most of them in the military inventory are used on armored vehicles for antipersonnel and attack against opposing forces vehicles.
As far as taking down a plane, a .30 caliber hunting arm of semi-automatic type bought at Wal-Mart would be just about as effective as anything you can obtain in the .50 BMG cartridge. and loads more cost effective.
The intent is to demonize a large caliber gun, then the next smallest until the lowly BB gun is confiscated from Cub Scouts.
.50 BMG refers to the cartridge, at least in popular usage. John Browning developed several machine guns in various calibers; the fielded, .50 caliber version is the M2.