Counterfeit
The most obvious comparison for counterfeit is moneylender (well, maybe throne room or procession, but it obviously plays way differently from them). Indeed, let's look at this. There are some type considerations (throne room, conspirator, venture, bank), but generally the differences in terms of copper trashing are these:
Moneylender is terminal. Counterfeit can't even be drawn dead.
Counterfeit gives a +buy.
Moneylender costs $4. Counterfeit costs $5.
Now, moneylender is a sort of mediocre 4-cost. It's not really a bad card, but it's not strong by any means. This is definitely better to have, so it rightly costs 5+, but you would think that it would not be too swell, given that the 4-to-5 price jump is the most important in the game. But that would only be true if Counterfeit could only be used as a moneylender. Of course, it's quite a bit better than that.
The next comparison I want to look at is salvager. Later in the game, counterfeit is very similar to a slightly better salvager-for-only-treasures. Counterfeiting a silver gives 5, whereas salvaging would only net 3. On gold, it's 7 vs 6. Platinum - 11 instead of 9. Of course, you can't trash actions or victory cards (or curses or hovels) with counterfeit. But again, it's a treasure instead of a terminal action, and it of course does MUCH better at copper-trashing than salvager.
There's of course some other uses for counterfeit, and some tactical decisions. Specialty treasures can give you double benefit. You can get the double-potion you want to end a vineyards game, or grab the last of the scrying pools or what have you. You can double Horn of Plenty to get two things, particularly nice if gaining victory cards (just remember that your HoP won't be able to count itself on the second play). It can make zany amounts of cash with bank. It just gives you lots of free coin out of your spoils, since you'd be losing them anyway (and they aren't even trashed, since they're already back in the supply!)
To follow-up on this, one draw-back of the card is that you don't get the in-play benefits of several treasures - royal seal and talisman are all less-than stellar as overpriced silver and copper, respectively.
Also, you can generally play counterfeit as copper-with-plus-buy. While this is not very good, it can be useful in some situations (where you've gotten the card for something else). Most often, this will come up when you're going for basically a treasureless deck - you still get a little benefit from your counterfeits, unlike what you'd have from moneylender or spice merchant. Occasionally, you also just won't want to trash something else. Maybe you're playing in a slog and want coppers (unlikely, given that you've got a counterfeit, but maybe a duchy was swindled into it and there aren't any swindlers left in your opponent's deck or the supply). This is a slightly better copper. More likely, trashing a silver or gold won't let you buy anything better, and you still want them for later. You generally shouldn't be very afraid to trash those cards, but there are little segments of time in some games where this will fit. The flexibility is nice. And because there's a "may" thrown in, you never *have* to trash anything.
Of course, there's the question of 'what happens if you counterfeit a counterfeit'. Fortunately, it's really straightforward - you play the first one, giving you a coin and an extra buy, then you play the second one twice - this will give you 2 more coins and 2 more buys and the opportunity to trash and double two more treasures from your hand. You have to trash the second one, though, so this is *usually* not the greatest option, unless you *really* need the extra $1 or extra 1 buy - the exact same options of other treasures to trash are available either way.
Most normally, the flexibility of the different modes makes counterfeit a quite strong card. The obvious use for it is in engines, where it trashes coppers out of your way early, gives +buy, and pulls in a few big cash turns later. This is pretty much just what an engine likes, so it's obviously a pretty good card here. However, its real strength (I think) is in big money decks. Yup, money decks. Particularly those where the key non-treasure card costs something like $4, and particularly when we're talking terminal draw BM. Smithy, eat your heart out. Why? Well, usually engines have things they want to do on 5. Most of them will want one or two of these, but it won't be *that* big a bump. It is, however, a pretty huge bump over the silver that a lot of BM decks have. The trick is in trashing the right thing, and for that, you just need a very good feel of when you need to aggressively green, and when you still need to be building.
The card's only real weaknesses are that it costs 5, and in slogs. It just doesn't do much at all for you in slogs, and while you often want to do something else on 5 anyway, this is even moreso the case in slog-land.
Works with:
Big Money
Engines
Most any kind of normal deck, where the first one is most usually preferable to gold
Conflicts with:
Junking attacks
Hoard (sort of - the extra golds to go through are very nice)
Royal Seal
Quarry (sort of - this still gives +buy and thinning, not bad for actions)
Talisman
A few power 5s