Pick Your Poison
Everyone has their own playstyle. Some like nothing more than to grind for experience and level up. The people who prefer this style tend to have multiple maximum-level characters, often on different servers, and frequently some of the same class. Leveling a character can take anywhere from 100 to 1000 hours of dedicated play time.
Some players like to farm gold or play the auction houses to get rich. In a lot of MMORPGs, the player with the biggest (in game) bank account has the biggest advantage, allowing him to purchase the very best gear and items. On large servers, a simple auction house for buying and selling items can become something of a miniature stock market, with players spending all day watching and recording price fluctuation, buying low and selling high. If that isn't their style, then they can spend hours on end out in the open world killing monsters for the money they drop, or farming items to sell.
Some players like to do nothing but fight other players in PvP (player vs player) for bragging rights or some special item. A player can become obsessed with being the best, and the most feared in PvP. In some games, PvP also earns experience points to help level up. In others, it earns you points that you can use to buy gear to make your character even better at PvP. One can literally spend 100+ hours doing PvP in order to earn enough points to have the best gear available.
Other MMORPG players only live to be able to say that they took down the biggest boss in the game. Many MMORPGs focus on an aspect called PvE (player versus environment), where groups of players go into dungeons or areas in the open world where there is a very powerful monster that it takes a very large group to kill. Killing these big bosses earns, not only money, gear, and items, but also fame, glory, and bragging rights. In games like World of Warcraft, where these monsters are a huge part of the game and storyline, many players don't feel satisfied until they kill the biggest boss in the game. Unfortunately, getting to that point can take months, or even years, and by that point new, bigger, badder monsters are just around the corner.
On top of these broad categories of things to do, another seemingly less important big feature in many MMORPGs: Achievments. What exactly are achievements? They are small tasks, such as traveling to different places on the map, killing a large amount of a certain type of monster, doing things while not wearing any armor, and learning some random tidbit of storyline from an NPC (non-player character) out in the middle of nowhere. That is only a few examples of the thousands of achievements that can be found in games such as Warhammer Online: Age of Reckoning, World of Warcraft, and Star Wars Galaxies, to name a few. In some games, accomplishing these small tasks unlocks titles and brags for your character to flaunt. Other times it is vanity items, that serve no other purpose than to look good and show off. In others, it is simply a list available when someone inspects your character. Regardless of the reward, players easily become obsessed with finding every achievement, adding a further layer to the addictiveness of MMORPGs.