When someone says that they read fantasy, what they typically mean is that they read high, low and/or epic fantasy. These three have some common denominators (though the rules may be bent for some of the other subgenres): The story is set in something akin to mediaeval times or the dark ages, there is an element of magic involved, there is a main hero, heroin or band of heroes leading the storyline.
In a High Fantasy setting, you typically get a lot of magic and magical races. Think Lord of the Rings (JRR Tolkien), The Sword of Truth (Terry Goodkind) and the Diskworld books (Terry Pratchett). There are magical races everywhere and almost every second character is either of a magical race or possesses some kind of magical item or ability. Magic and magical races often even determine the culture and societal dynamics of the worlds the stories happen in. Dragons might be overlords or a slave race. Orcs may be a powerful ally in the subterranean reaches. Elves and men may have formed an alliance (when don’t they???).
In a Low Fantasy (or Hedge Fantasy) setting, magic and magical races are rare. These books often use the one magical being as the main character or have a magical object at the core of the plot…because the magic is so rare. These worlds are typically more ‘realistic’ and often based more closely on actual mediaeval day-to-day realities. Some Low Fantasy stories even follow the mundane lives of unimportant people (though they often have important roles in the bigger picture). The Farseer Trilogy (Robin Hobb) and A Song of Ice and Fire (George R Martin) are good examples.
In an Epic Fantasy story, the level of existence is often upped to a new level. Your main character may be a demi-god or a similarly powerful creature. The story will revolve around a battle of the gods or kings and other great rulers shaping the very world and reality of others. Some examples are: The Lord of the Rings (JRR Tolkien), The Belgariad and The Malorean (David and Leigh Eddings) and The Rose of the Prophet (Margaret Weis and Tracy Hickman). Naturally, Epic Fantasy and High Fantasy often get mushed together with great success.
Something that may be important to remember is: Each story is a unique blend of different subgenres. No subgenre stands purely alone. Only successful blends in subgenre ever make it big out there, though.