Lilfox - "well drinks" generally refer to the alcohols in the well, aka the speed rack. These are house liquors (non name brands, or budget brands). There are tiers of alcohol - well, bottom shelf and top shelf, with the latter two physically located behind the bartender, the well or speed rack bottles located in a lower rack in front of them, for quick and easy access. So if you can see the bottle behind the bartender, its not a well bottle. Often juices are also kept in the well, if not in the gun.
To order a well drink, you are limited to the bare basics: (usually) rum, whiskey, vodka, tequila, and gin. No fancy flavor variations like citrus vodka, or pineapple rum, just plain ole vodka, plain ole rum. And they aren't ordered by brand, just by type.
vodka & orange juice = well drink
Grey Goose & orange juice = NOT a well drink
whiskey sour = well drink
Chivas sour = NOT a well drink
rum & Coke - well drink
Captain Morgan's & coke = NOT a well drink
gin & tonic = well drink
Tanqueray & tonic = not a well drink
etc, etc, etc
Some places will allow well cocktails, like a tequila sunrise, or a martini, but its still going to be made with the speed rack liquors not some fancy 100% blue agave tequila or a top notch gin or vodka. But for the most part the point is the drinks are reduced price because they are quick and easy to make as well as using only basic ingredients.