A hard-on produced or facilitated by Viagra, or any other such prescription drug for "erectile dysfunction." Viagr-ons are different from naturally-occurring erections in that they are more easily attained and don't require the same amount of genuine sexual stimulation. Also, they tend to last longer and throb more intensely.

when you're hungover as shit after a spending a night drunk as a skunk

Having an erection or hard-on caused by taking Viagra.
Could last several hours.