Never. (Note: new edits. As many more expansions have been released since I first wrote this answer, there's a lot more exceptions.)
Well, almost never. If you have to ask, the answer is "never". Don't buy copper unless you've got a specific strategy that will do well with it. Some new players have a strong urge to use all their buys, or at least to buy something every turn, but buying nothing is better than buying Copper except for in hedge cases.
OP mentions the Gardens exception in the question; other cards that can be used effectively with Copper include Goons and Forager (thanks @John Robertson). As @Gendolkari comments, if your target buys aren't too expensive (say $5 or less), then Coppers can help you get there; this applies to Duke/Duchy, Gardens, Ill-Gotten Gains, etc.
I've seen new players make the mistake of thinking more money is better, so they'll buy a Copper with a leftover buy. This is obviously bad because it's the money in your typical hand is what matters, not the money in your deck. With 5 cards in your hand, adding Coppers is going to
reduce your buying power unless your deck is really bloated (maybe with Curses and Ruins). Players quickly learn to hate a hand full of Coppers when they want to buy a Province, and stop making this mistake.
Just think, when you draw a card, are you thinking please, please, let this be a Copper? If so, then maybe you should be buying Copper - if you'd prefer having a Copper to what you do draw (maybe because of all the Curses/Ruins in your deck). More often, you are hoping for something better than a Copper and you are disappointed if you draw a Copper - this is your cue to not buy Copper.
It's much more common for somewhat more experienced players to overestimate how good a Copper-using card is. Examples below: Moneylender, Coppersmith, Bishop, Mine.
Moneylender can be a great card, but some players will think it's a great card because of the $3: Moneylender shines in the absence of other trashers because it cleans your deck of Copper. Especially since Moneylender is a terminal action, buying several Moneylenders and then buying Coppers to feed them is not a viable strategy by itself. It requires 2 cards and an action to get $3; if you had two Silvers instead of Money Lender and a Copper you'd have an extra $1 and an extra action to use. Even Silver + Money Lender can be better than Copper + Money Lender if you've got a different action card worth playing. Spice Merchant and Miser are very similar.
With Coppersmith, there are decks in which it can work very well (@BlairHippo brings up Apothecary/Coppersmith), but it takes more than just Coppersmith. You need a lot of card draw for Coppersmith to work well, but card draw helps most everything so there are often more viable strategies on the board.
Bishop can convert Copper to VP, which is great. But anything costing $2 or more gets you more VP. With extra buys, you'd be better off buying Estates than buying Coppers to feed to a Bishop.
Mine is also like Money Lender. What makes Mine a good action is that it gets rid of bad treasure, replacing it better treasure. If you then go and buy more bad treasure to feed to a Mine, you're missing the point.
Counting House/Travelling Fair is a strategy that is copper-centric, meaning that copper is very good.
Of course, if you have a bunch of activated Cities (lots of actions and cards) and a few Goons (lots of buys, rewards for buying), go ahead and buy some Coppers and Moneylenders/Coppersmiths/Spice Merchants etc., but if you've got a good Goon engine going your probably don't need much advice.