A deck building game, AKA a deckbuilder, is any game that is based primarily on the mechanic of deck building. This is an extremely popular subgenre of games, and once you get the basic gameplay down it makes it fairly easy to pick up other games that share this mechanic. Deckbuilders are generally referred to as board games, despite the fact that they mostly use cards, not boards and tokens.
Most deck building games give each player a personal starter deck of (crappy) cards to start with, and players use these cards or other game events to upgrade this deck. This is almost universally achieved by buying cards from a shared bank.
A generic deckbuilder might have a starting deck of ten cards, consisting of 'money' cards and 'dud' cards. Each money card represents one 'gold coin'. A player shuffles these cards and draws five cards, and uses the 'gold' they just drew to buy a card from the central bank. The card they purchased and the cards they spent all go into their own personal discard pile, and will be shuffled and reused as soon as their drawing deck runs out.
In this generic game, we might expect purchasable cards to include cards worth two or three gold, rather than just one; cards that allow you to banish 'dud' cards from your deck; cards that let you draw extra cards; cards that give you victory points; and etc.
Most games have some sort of attack and/or defense system built in. In Ascension one type of card that appears in the center row represents monsters, and players must add attack cards into their hands to beat them; in Star Realms attack cards are spent attacking the other player, and so players also have to spend part of their budget on defense cards.
Deck building games include Dominion, Ascension, Star Realms, Legendary, Friday, Mystic Vale, DC Comics Deck-Building Game, and The Big Book of Madness, among many others. If you are interested trying a deckbuilder, Ascension: Apprentice Edition and Star Realms provide cheap and easy starter games. Dominion is a classic and a popular favorite, and while it is a little more expensive to buy the base game it is a good starter game, as the central bank of cards is not randomized, allowing for much easier long-term planning.
There are also deck-builder-like games that use dice, such as Dicemasters and Quarriors, and games that use tokens, such as Puzzle Strike; these games replace the decks with bags from which players randomly draw their dice or tokens.
Deck building games may be considered a subcategory of pool building games, in which a small starting resource pool is built up into a larger pool (familiar from traditional games such as Monopoly); deck building games add the mechanic of giving you access to only a randomized segment of your overall pool each turn.