Deck-building is a popular game mechanic in modern board games and card games. You start with a small deck of crappy cards and use these to gain better cards. Some of the cards you earn will allow you to remove the less desirable cards from your deck, or add value to other cards. Each turn you draw a new hand from your deck, play them as you see fit, and discard them. Once you have gone through your deck, you reshuffle your discard pile and start again. By the end of the game you have built yourself a deck optimized for your chosen strategy, but so have your opponents. You count up your points and reset all decks back to zero for the next game.
The starter deck generally consists of cards of either one or zero value; usually ten cards, of which the player will usually draw 5 each hand. The cards will usually have something equivalent to 'money' and 'attack' -- although the DC Comics DeckBuilding Game combines these into a single stat. Other stats may come into play later in the game, most particularly victory points, which are usually treated as an independent metric.
The good cards are all collected into a central marketplace; generally an unlimited set of low power cards and a limited set (usually five) of nicer cards. These cards are replaced from the central deck upon purchase.
Most, perhaps all, deck-builders have various factions, classes, or item sets that synergize or interact in interesting ways. This is one of the primary reasons that deck-builders are addictive; although it will take only a couple of games for you to know the cards, exploring the different combinations and their benefits takes much longer.
Most deck-builders have expansion sets available for purchase, but these are always structured (that is, you know exactly what you are getting, meaning that you do not have the collectible card game aspect), and all players have equal access to all cards.
There are quite a few deck-builders out there, and I won't try to list them all. Below are some of the more popular ones, and I will add in any others that I find reviewed on E2. However, if you are looking for a good beginner's deck-builder, there are two obvious choices: Ascension is very popular, easily available, and not too expensive -- and it is perhaps the game that best mirrors the average deck-builder; and Star Realms is perhaps a bit more intuitive in its game play, as it replaces the murky victory point system with a simple 'attack you opponent until dead' system that is quite easy to track. It is also cheaper than many games, and has a number of expansion packs available.
If you are curious about these games, many game shops have demo copies available try out, and some even have board game events where many people who enjoy this sort of game will be available to teach you the basics.