(http://wiki.dominionstrategy.com/images/e/ea/Prince.jpg)
Prince is the shiny new card that got us all excited for a while until the new expansion was announced. The eventual effect – being able to play a card every turn without using an action or card slot – is super strong, but it's a long way until you get there.
Prince of what?
Let's start with the easy part – determining which cards you want Princed. Usually, once you bought a Prince, you want to set it aside as soon as possible. But if you have two potential targets, deciding which one to Prince might be difficult. Plus, if you see a kingdom full of cards you don't want Princed, you probably shouldn't go for Prince. I'll present a list of very bad to very good Prince targets.
The completely useless
Durations, one-shots
These cards just don't work with Prince, for technical reasons.
The very bad
Ambassador, Baron, draw-to-X, Cursers/Looters, trashers
Cards you probably don't want to Prince even if you can. With these, it's often better to Prince a shuffle later. This includes cards that require you to have certain cards in hand to be good, like trashers (require junk cards), but also cards you want to play a lot of early on and that lose value quickly (junkers).
The rather bad
Throne Room, Villages, Menagerie, Reactions
Cards you will probably Prince if you draw your Prince with nothing else, but where the effect just isn't great. Princing reactions means losing the reaction part, if you Prince a Village, you will usually end up with lots of unused actions. Throne Room can end up doing nothing, or even throning a bad card [side note: if you Prince Throne Room and throne a duration with it, you lose the Princing].
The rather good
Cantrips, trashing attacks, gainers, +buy, sifters, +coins
Cards that are almost always nice princed. Cantrips help you avoid terminal collision AND let you start with bigger hand size, which is already two plusses.
The really good
+cards, Discard attacks, Monument, Tournament, Scheme, Prizes
These kinds of cards are exceptionally good. Starting with a bigger handsize each turn is huge. A princed discard is sort of the inverse effect - opponents start with lower hand size each turn. Monument gives 1 VP per turn. Scheme effectively lets you play a more expensive card each turn. On tournament and Prizes, see below.
Note that this list does not imply that you should go for Prince whenever good Princable cards are around. The decision to make if e.g. Tournament is around is much harder than this. The following part tries to give some guidelines for deciding when to go for it.
When to go for Prince
First and foremost, Prince is an engine card. It has no place in big money games or slogs, simply because – similar to Throne Room/King's Court – you need to line it up with an action to get anything out of it. There are rare circumstances where you might want to use it even in non-engine decks, but the general advice is, ignore it if you're not building an engine.
Even if you are going for the engine, consider carefully if you really want Prince. The main problem about Prince is that it is slow. That Prince could have been a Province, and then you don't get the effect immediately, but instead need to sacrifice on action to set aside the card you want. That puts you behind a Province and roughly half a turn. If you could buy a single Province per turn before and now can buy two Provinces a turn, that may have been worth it; but more often than not, the benefit simply won't be big enough.
The following list gives scenarios where Prince can be good.
1. Colony games
There are two reasons why Prince is better in Colony games: First, Prince is made for long games, and Colony games tend to go longer. The more turns you have, the more will you be able to play your Princed cards. Second, the competition for Prince is weaker. In a Colony game, you usually don't want to go for Provinces except in the endgame, so if you hit $8, you can grab a Prince without that much opportunity cost.
2. Alt-VP
Same as above – if you don't want those Provinces anyway, ignoring them becomes easier. This point comes with a caveat though: Most Alt-VP cards (Gardens, Duke, Feodum) support decks that don't want Prince in it. But Prince can be great with Fairgrounds and Vineyard, and there's always the odd engine-into-Duchy/Duke game. The VP token cards also fall in this category: Prince allows you to build better Golden Decks for Bishop, a Prince of Monuments nets you 1 VP per turn, And Prince an help play more Goons per turn.
3. Prince while ahead
If you already have a decent advantage, you may be able to cope with skipping a Province buy. Prince can give your deck additional reliability and prevent you from losing to unlucky draws.
4. 2-card combos
This is a bit of a niche use for Prince, but can be quite good. Some cards combo nicely in theory, but it just isn't worthwile to set them up, because e.g. the one you'd need to play first is terminal. Prince can help with that. Prince a Navigator and your Herald engine will flow smoothly. Prince a Scavenger to get the card you want on top of your deck.
5. Rare components
This is definitely the scenario where Prince shines most. Sometimes, the engine is just not quite powerful enough. There are awesome eninge components, but the only Village is Necropolis? Just crown a Prince of Necropoles (is that the correct plural form?), and you'll start with three actions each turn. Similarly, a Prince of Crossroads can make your engine work.
While +action is what you need to make the engine work most often, there are also other possibilities: Maybe you can't guarantee drawing a +buy every turn – set it aside. Princed attacks are also quite nice.
When to play Prince
Usually, you want to Prince your cards as soon as possible. If you have Prince with a decent card in hand, don't wait for a better opportunity! Another mistake that is often made is this: you have Prince and Wishing Well in hand. Unless you can be pretty certain that you can draw a better action, don't play the Wishing Well! It's tempting to try and get more out of your turn, but if your Prince ends up dead, you'll curse your recklessness.
Countering Prince
Not much needs to be said here. If your opponent goes for Prince and you don't, your task is to end the game before his Prince pays off too much. Piledrive these Provinces, go for the three-pile. The longer the game, the better for the Prince player.
Specific interactions
In this last part, I'd like to pick a few special cards that have an interesting interaction with Prince.
Prince and Tournament
Prince and Tournament have a love-hate relationship. You need Provinces to activate your Tournaments, and every Prince could have been a Province. On the other hand, once you've Princed a Tournament, every hand with a Province in it wins. And, even better, once you've got those Prizes, you can Prince them and quickly get your opponent to resign. Getting Province with your first $8 and Prince with your second isn't a bad strategy on many Tournament boards.
Prince and Black Market
No, not what you think... Princing a Black Market is of course horrible. But what Prince really likes is the Black Market deck. Often you get a good card from the Black Market, but it's not enough because you can only ever get one copy. With Prince, you can play that card every turn.
Prince and cost reducers
Prince loves cost reducers because they give the opportunity to set aside better cards. After two Highways, you can set aside a Goons. Just don't spend too much time on such neat tricks. It's often better to set aside Prince this shuffle with a boring 4-cost than to wait another shuffle in hopes of getting that Prince of Hunting Grounds.
One warning: If you Prince a Bridge, keep in mind the effect that has on your TfB cards. You definitely look stupid when you start Upgrading your Coppers into Estates.
Prince of Hermits
This can be great if the kingdom allows for it. If you don't buy anything a turn, Prince sets aside Hermit before it can trash itself. That means, if you can cobble together an engine that doesn't need to buy anything, the Prince of Hermits can give you a constant supply of Madmen.
Other interactions
Outpost: more turns mean more Princed card plays. And Prince is often enough to make those Outpost turns as good as normal turns.
Quarry: Lowers the bar for Prince. Now it doesn't compete with Province anymore.
Swindler: Just like Peddler, the presence of Prince makes Swindler games even swingier than usual. If you manage to turn Province into Prince late-game, this can be a game-changer. And unlike Peddler, Princes will very rarely run out towards the endgame.
Conclusion
Prince is a unique card, but don't let that distract you. It's not that good. It has its uses, but you usually shouldn't just buy it without good reason.