Most people know that 'being the Village Idiot" is something to avoid. You don't want to open with a village because no matter how many actions you need, there's little point in picking up a village on turn one when you don't even have two terminals to play it with. Pretty simple, right?
Well, the story goes a bit deeper. Consider how a lot of players look at smithy early on. Good card. I can draw a lot with it. But man, I draw stuff dead with it. Especially other smithies. I wish there were some way around that. Hey, village! And so they then proceed to build big smithy/village decks. Lots of villages, lots of smithies, drawing like your whole deck every turn. And then they lose! The tragedy is, for a good while, players will often then chalk these losses up to bad luck. But actually, in a kingdom where smithy, village, and 'the basics' are the only cards, you know what the optimum number of villages is? 0! (and that's not the mathematical zero factorial, which would be one, but just plain zero). Huh?
The thing is, village isn't a bad card. Like basically all other at-least-a-cantrips, in the vast majority of decks, if you gave me the choice of having a village or not, I'd have it. I almost always give the village to myself. So why then, is it better to not buy the villages here? The answer is opportunity cost. Basically every time you buy a village, you've wasted at least $3 and a buy; you could have bought, if nothing else, silver. And most decks need silver early on to ramp up to buying 5s and 6s and eventually provinces.
Let's take a closer look at the village/smithy deck. Of course one of the other biggest problems here is that you can draw the villages dead. To reliably be able to chain stuff up, you need to make sure you have a village in your original 5 card hand, so you need more villages than smithies. So to draw most all your deck, you'll need something like 5 villages and 3 smithies. Let's say you've thrown a couple silver in 'cause you realize you need them. Well shoot, now you need a couple more smithies, which means a couple more villages. So that's 7 villages, 4 smithies, 2 silvers... that took 13 turns to set up, and you can still start without the right combination sometimes. Big money baselines to 4 provinces in 17 turns. With smithies, cut that to 14. Our strategy, it seems, is just too slow then.
So the next logical progression is to ask about if you have good reasons to draw a big hand every turn with actions to spare. Good question! There are definitely some good terminals out there to help your village/smithy engine. How about Merchant ship? That's good. It speeds your deck up - maybe you can skip some of the money (it's pretty risky). But it doesn't really get you into the speed you need. Plus, why not just buy money and then ramp up to merchant ship? Do you really need to play it every turn? No. So what cards do you want to be able to play every turn? Most of them are attacks. The problem is, basically every attack does more damage to an engine like village/smithy than it does to money. Curse-givers especially fill your deck with muck and you'll only be able to get the key cards in the right order very rarely.
The same story goes for most other cards - you should just buy those cards without village support. Yes, they can have terminal collision. But you would need both those cards WITH the village to avoid that, and that's pretty unlikely. And if you correctly gauge how many of the terminal you buy, you won't get that collision too often anyway.
Okay, so village isn't the uber-card we all thought it was back when we got actions with every buy. But does it really suck so bad you should almost never get it? No, it can be good in a decent number of situations, like most other cards. What are those? First of, there are basically two cards, I think, that warrant heavy village strategies almost by themselves, if there's no other amazing thing in the kingdom: Torturer and Goons. Both of these cards share the property that they're really, really good if you can play multiple copies in a single turn, especially if you can do that reliably. Torturer chains can shut the opponent down cold. Goons stacks build up VP chips really quickly. There are two other cards that have this property, bridge and conspirator, and village works well with them too, but you really need to play a lot of bridges for htat to be super good, and with village as the only support for conspirator, it's going to be tough to hit the critical action mass to get them going very quickly and you won't be able to do all that much with it once you do. But with a little other support, village with these can be good too.
But the most common reason you want village is in the midgame. You already have a handful of silvers, and you've bought three or four good terminals and would like to buy more, but you're worried about terminal collision. Well, since village IS a cantrip plus, and since you've already got a pretty good amount of money, and these terminals are good enough that you'd really like to be able to play multiples sometimes (especially if they aren't card-draw), then village can be a very good fit in this deck.
Basically all of the above applies to many of the 'village plus' cards. I'm specifically thinking of worker's village and walled village. Shanty town is usually even worse if you want to use it as a village. But there are some caveats for some of the variants:
Bazaar is much better, as that $1 helps actually quite a lot. Of course, it also costs $5.
Mining village is much better since you can (and pretty aggressively should) trash it for a one-time silver
Native village shouldn't really be thought of in this class at all, as it doesn't draw the card in your hand directly. Thus it's much worse at being a prototypical village, but better for a lot of combo decks. But again, plays differently.
Finally, there's fishing village. Fishing Village is really another beast entirely, generally a much better card, and deserves its own article, which will come at some point.
Admin edit: added whitespace between paragraphs for readability