Being an MMORPG gamer, there is no "best game" to play. Aside from the core concept of MMORPGs themselves all games are different. If you haven't played any RPGs at all, the best way to choose a game is to actually play few of them. Most MMORPGs require you to spend some cash/month.

Also there a few kinds of MMORPGs: desktop(ordinary games) and browser-based-games.

The current hit is probably World of Warcarft(WoW). Ofcourse there are other notable games like LineAge2, Eve and many more.

Browser-based games are playing in a browser, mostly Firefox. A big name is Runescape, around 10 millions of registered players.

My advice for you is to try and play Runescape as it is free for the most part and you can get a feeling of what is a MMORPG. After that you can decide to jump to WoW if you are still interested.

Runescape is probably the best free MMORPG. It has a large variety of skills, and lots of hours of play. The downside is that members get a big advantage, and leveling up can take a really long time.

Sherwood is another fun MMORPG. I had some problems saving my progress with it, but you should give it a try. Sherwood is completely free. Basically, you just go around fighting monsters, and leveling up.

World of Warcraft is a great game too, but you have to pay every month.

Guild Wars is another good MMORPG. You have to buy the game, and then you can play it for free online as much as you want. You can find Guild Wars for cheap off of Ebay.

The biggest downside when paying every month for a game like Runescape or WOW is this:
You will feel like all your time in the game is precious. In these games, you will feel rushed and want to make the most out of your time. If you play Runescape for free, you won't feel bad if you waste some time talking or exploring the world.