Foil cards are cards that have been adorned with a special border based on the color of the collection. For example, Foil cherry cards will show a faint image of cherries, while foil lavender cards will show a faint image of flowers, and so on, for each color family.
Players value Foil cards more because they are rarer than regular cards, and because collecting them can generate special Foil Feat badges when you complete a collection with all Foil cards, or when you’ve collected a foil card from each color family.
Players have no control over whether or not their card is foiled; Foil cards are created randomly through normal course of play in PackRat; sometimes you’ll just get lucky and draw a Foil card, or buy one, but you cannot see that it’s a Foil card until after it is purchased, unless it's being sold in the auction market, where already Foiled cards are sometimes offered for sale.
It's important to note that XL players have increased odds of foiling cards compared to standard players, and these odds are based on player type, NOT what market the cards were purchased from. The odds of foiling cards are highest if purchased from a ticket market.
Furthermore, PackRat sometimes releases special "decal" collections whose cards, when foiled for an XL player, are overlaid with a special decal based on the particular card in the collection. Standard players will only see the regular Foil, but are are free to trade for (or steal) such cards.
For example, in the collection 'It Must Be Bunnies,' when the 'Presto' card foils, it gains both the blueberries, AND a special decal that adds a magician's hat, cape, and wand. So while an unfoiled Presto card looks like this:
The XL decal version of the card looks like this:
Foil cards can also be generated when a recipe is made, and your odds of making a foiled recipe card increase based on whether the component cards are foiled. For example, you have a very high probability of foiling a recipe card if the three component cards are all foiled; this probability is less if you have 2/3 of the component cards foiled, and less still if you have only 1, or none, foiled.
After vaulting Foil cards, you can easily see all of them in your Vault, regardless of collection, by clicking Foil Cards in the drop-down menu: