Source: Yugioh Wikia
Monster Cards represent monsters that players battle against each other or directly against either player during the Battle Phase. Monster cards are differentiated by their names, Type, Attribute, ATK, DEF, Level. Furthermore, there are Normal Monsters, which are monsters that lack effects and aren't any of the other kinds of monsters, in addition to Effect Monsters, Ritual Monsters, Fusion Monsters, Synchro Monsters and Token Monsters. Note that Effect Monsters cannot also be Normal Monsters, but they can be Ritual Monsters, Fusion Monsters or Synchro Monsters. Also note that Ritual Monsters, Fusion Monsters and Synchro Monsters can lack effects, making them unable to be Effect Monsters, but they are not considered Normal Monsters.
During play, monster cards can be played in either face-up Attack Position or face-down Defense Position. The importance of monster cards lies in their effects (which usually activate during their controller's Main Phase) and their ATK and DEF values during the Battle Phase (or more specifically, the Damage Step).
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. e.g. a "Samurai" monster is a monster with "Samurai" in its card name.
Here's a List of monster types.
For a list of Normal Monsters.
For a list of Effect Monsters.
For a list of Ritual Monsters.
For a list of Synchro Monsters.
For a list of Fusion Monsters.