Typically, each player will draw one card on each of their own turns during the Draw Step. In a Duel (2-player game), the player who goes first does not draw on their first draw step. In multi-player games, the first player to go does draw on their first draw step.
Many cards can cause you to draw more cards, either during your draw step (e.g. Howling Mine, Font of Mythos) or at other times during a turn (e.g. Ancestral Recall, Courier's Capsule). The ability on a card to allow you to draw a card, especially on one shot cards such as instants and sorceries, is colloquially known as a "cantrip". This usage is most often utilized when the card has an ability already and the "Draw a card" ability is the other ability.
If a player would draw from their library but can not do so because their library is empty they lose the game.