When you're willing to pay to play a game online, you have lots of great choices. However, if you're not willing to pay for an online game, finding a good game that is also free to play can be a bit like finding a needle in a haystack.
The first and most obvious step will be to google "free online RPGs". This will spit out a lot of...Well, bad games. Your best bet is to look for a site that has a list for free to play online roleplaying games. Most likely they will have a function that allows people to rate them.
Ratings on those types of sites are generally pretty reliable, however; each person if different. If you want an roleplaying game with fluffy bunnies and unicorns, that probably won't be rated so highly by the internet community.
A second step is to make friends in online roleplaying games that you play or on forums about online gaming. When it comes to free games, they can't really afford to advertise their game so a lot of the advertising is done with word of mouth. Chances are if you're into online roleplaying games, you have some friends that are too. So ask your friends about free alternatives to paid roleplaying games. Really one of the funniest ways to enjoy any game is with a friend or six.
Look for older games. If a game has been online awhile and has always been free to play, chances are, it's pretty freakin' good. However, be wary of games that were once pay-to-play and that are now free-to-play. They've obviously downgraded because they were hemorrhaging subscribers, so thus something might be wrong with that game. Like I said though, some games just aren't for everyone.
While free is awesome, I still encourage people to try free trials of paid games. If the game is popular still and requires people to pay a subscription fee, there must be something that keeps people coming back. Though always try them before you buy them, I hate buying games that I end up hating.
Look for open betas. Open betas are great, because companies don't want to pay actual testers or want public opinion. Open betas are a great way to preview a new game.
When playing a free game, be prepared for some adolescent shenanigans. By adolescent shenanigans, I really mean young kids with foul mouths. This is really one of the biggest downfalls of online gaming, because it's free kids can get at it really easily. They don't have to beg their parents for money, they just have to have a computer with internet. You know how kids are, they think they're the best! So, yeah, be ready for that.
Be flexible. If you are looking for something specific, why not branch out? If it's free to play what's the harm in trying something new? You may just find out the kind of game your were looking for is not what you really wanted.
Some suggestions (I won't link though, google them):
Aoin (was pay-to-play, is now free)
Star Wars: The Old Republic (was pay-to-play, is soon to be free with paid options)
League of Legends (was pay-to-play, is not free)
Guild Wars and soon Guild Wars 2 (buy like a normal game, no subscription fees)