"Class 3" is really a misnomer - its become a common language designation for any weapon regulated under the National Firearms Act (NFA). Weapons regulated under the NFA include machineguns, supressors (silencers), Short barreled rifles (having a barrel less than 16" or an overall length less than 26"), short barreled shotguns (having a barrel less than 18"), Destructive Devices (grenade launchers, explosive devices, weapons with a rifled bore of more than .50 with no sporting purpose), "Any Other Weapons" (a catch-all category including guns disguised as other items like cameras, flashlights, etc., handguns with a vertical grip, not designed to be fired with one hand).
The NFA is premised on the taxing authority of Congress, and as such the NFA and its implementing regulatins are found in the tax code. All transfers of NFA weapons are subject to a transfer tax (hence the relation to tax).
To buy an NFA weapon, it needs to be legal in your state, and there is a bit of federal paperwork involved; pictures, fingerprints, a "sign-off" by your local law enforcement, and of course the check for the transfer tax ($200 for each transfer, except for AOWs which are only $5).