General
A card is drawn automatically at the start of a player's turn. Upon starting a battle, each player is presented with three cards, and may choose to replace any number of their cards for a chance at different ones.
Ability
Several abilities/spells, such as 
Arcane Intellect, Wrath or Flare, allow you to draw cards when played.
Minions
Some minions have a draw card mechanism activated upon either entering the battlefield (Battlecry), dying (Deathrattle) or simply by attacking. If a minion gets silenced they lose the draw card perk.
"Strategy"
Card Advantage is a very important part of card games. What this means is when you play a Loot Hoarder, if you attack your opponent's Knife Juggler, both minions will be destroyed. This leaves you with drawing a card, and your opponent with nothing.
It's very important in gameplay to make profitable trades (eg. trading your Patient Assassin for their Molten Giant) but another method of gaining advantage is to use multiple cards (eg, using Bloodfen Raptor to break Sunwalker's divine shield, THEN attack with Patient Assassin) which can only be done profitably if you have more cards in hand, and on board, than your opponent.