Steam Trading Cards are used to craft Badges, which can raise your Steam level and give you themed goodies depending on the badge you craft.
Most cards are earned by playing games that feature card support, but some are special for particular events.
The cards related to the Winter 2013 sale are used for crafting the Holiday Sale 2013 badge, which drops special in-game items for several different free-to-play games.
When you vote for the Community Choice game on the main page 3 times, you'll earn one random card towards this badge. It takes a set of 10 unique cards to craft it, and you can craft it multiple times. Each time you craft the badge, you'll get another in-game item from a selection of in-game items for certain free-to-play games.
There is a minimum Steam level in order to earn cards, and so if you are too low-level you'll end up not earning any cards from your votes. If you manage to get your Steam level up during the sale, you'll be retroactively rewarded the cards you would have earned from voting.
However, unlike most other cards, the Holiday Sale 2013 cards are going to evaporate at the end of the sale, so if you don't get your Steam level up before then, they will disappear.