Card advantage is one of the most important concepts to understand in Hearthstone, and relates principally to the idea of having more cards in-hand than your opponent does, while having some kind of equivalent presence on the board. Weighing up what you have in play with what you have in your hand - and factoring this against your opponent's own set of circumstances - is an important part of developing your strategy of when to trade minions and when to play more aggressively.
Here are some tips that you might find useful for gaining a deeper understanding of the importance of this Hearthstone mechanic, and thinking in ways that will lead you to gain the greatest advantage on the board now - and in the turns that lie ahead of you.
1. If you're designing your own deck, try to include the means to draw more cards through spells or minions. Most decks have fewer late-game minions than they do early-game creatures, and as the match goes on you're more likely to find yourself fielding multiple minions onto the board. If you can keep refilling your hand beyond the card you automatically receive each turn, you'll protect yourself further against board wipes or a snowballing threat from the opponent.
2. Aggressive decks typically require more card draw than Midrange or Control archetypes. With an aggro deck, you'll empty your hand much faster, and so be more vulnerable to board clears. If you're struggling to keep up with the opponent, look closely through your Hero-specific and Neutral cards to find ways of beefing up your minion flow.
If you're using spells as a means to gain more card advantage in Hearthstone, make sure they represent decent value. Sprint (especially when combined with Preparation) is a solid hand-filler, whereas Mind Vision merely spends one card to gain another. It may well give you a powerful card to play with, but you're not gaining card advantage. Consider also when to play your card-drawing spells - later in a match is almost always better so that you can compete efficiently on the board.
The dream scenario for gaining card advantage through efficient minion play is to find a way for each of your creatures to take down at least two of your opponent's. Look closely at the instant damage effects available to you in your hand, think about how they'd work with the minions you have out on the board, and look to push efficiency in your trading as hard as possible. Cards like Shattered Sun Cleric also often allow a weaker minion to punch above it's vanilla weight in order to take down more enemy minions.