In most games of Dominion, everyone has equal access to the same cards. There are, however, three cards that cause unique elements to come into play. There is only one copy of each Black Market card, one copy of each Tournament Prize, and one copy of each Knight. Sometimes, these unique cards can be extremely important because they can offer a potentially crippling asymmetry.
This is most apparent with unique cursing attacks. In a kingdom with no cursing attacks, Followers can be a game-winning Prize. Getting the only Witch out of the BM can be a huge boon. With other cursers, however, these situations become less important. Other unique cards may have their uses as well. If you're playing an engine with no +buy, Princess can really shine. Got Bridges? She's much less important. Getting an early Monument or Goons out of the BM can build you a nice pile of VP chips that your opponent has to match with green.
Knights are unique, but most have powers usually found elsewhere in the kingdom - +$, +cards, +action. One important possible exception to this is Dame Anna. If she's the only trasher on the board, it can be the only way to build an engine if that's what you're after. I had an opponent once forego other strong $5s to open with Dame Anna. Knights weren't great on their own, but Anna was the only was to trash anything on a board with good engine potential. Dame Molly can be useful for extra actions, if she's the only +2 actions available.
It's interesting to note that the widely-regarded booby Prizes, Diadem and Bag of Gold, are weak due in part to the fact that their effects aren't really unique. There are plenty of ways to get +$, and buying Gold is always possible given the cash. Similarly, cards without unique effects in the Black Market tend to be a bit weaker. The key is to find cards that are not only good, but that are even better if you're the only one who has it.
Overall, my point is that when unique cards are on the board, it's very important to identify which ones are key. Abilities unique to the kingdom (only +buy, only curser, only trasher, only VP tokens) can be game-changing, so don't pass up an opportunity to grab them if you can put them to use.