When you play a Monster Card, you can play it in one of two ways. Either face-up Attack Position, face-down Defense Position or in face-up Defense Position. Although, you cannot play a monster in face-up Defense Position to start with.
Monster Cards have two numbers on the bottom that usually matter, ATK and DEF. ATK only applies when in Attack Position, and DEF only applies when in Defense Position during the Battle Phase, but certain cards may still call upon either of these values no matter what position the monster is in. Monsters also have an Attribute, Level, and a Type. Skilled Manipulation of these variables will allow you to distort ATK and DEF to your advantage.
When a card's text reads a word or phrase in quotation marks followed by monster, it means a monster with the word or phrase in its name. eg. a "The Six Samurai" monster is a monster with "The Six Samurai" in its card name.