Anyone can take a character spreadsheet, fill out a set of numbers, pick a few cool sounding skills and write down cool names of some gear. In the end though, that side of tabletop RPGs is just function; the bones and organs of a set of rules.
It takes a special kind of player to Role-play.
Before I had ever filled out my first character sheet for Dungeons & Dragons, the only experience I had with RPGs was in video games. Whenever my tabletop-playing friends would go on about D & D and Shadowrun, I would usually answer with an apathetic "that's cool" before trying to change the subject. It was not until I finally played a game that I realized what they were all so excited about.
My first character was a Half-ling Bard named Theodore Reamus. It started with simple things: picking the best skills for optimal support and damage effect. Yet as I continued to develop him, I realized he was a lot more than a series of stats. He was a character just starting his epic journey.
For effect, I said he always wore tattered vagrant clothes and spoke in an Irish accent, despite the fact that the Feywilds are nowhere near Ireland. With the game master's permission, his instrument was "The Axe", a combination of a lute and two-handed axe. And whenever he cast his daily power "word of life", a chorus of woodland creatures came in to sing backup vocals on a version of "Circle of Life".
Now this game was more based on humor (the Arch-villain was a bartending necromancer from a village called Newcustle), so your own GM may not be as tolerant. Still, it illustrates what makes for good role-playing: you need to actually take up a role. This might sound easy, but it's something that both the player and the GM needs to remember.
First, the players need to remember that they're telling a story with the GM. If they want to just wield epic weapons and kill hordes of monster, they should just stick to World of Warcraft because they'll get just as much satisfaction from that (not that there is anything wrong with WOW). There is less of killing X number of gnolls or finding that one item being held by one random NPC. It's about being a character in a story.
For example, what class are you? A Paladin is guided by his morals and a deep-rooted sense of good. A Rogue will more than likely be out for himself or will be incredibly crude in manner and speech at times. Or you can choose to flaunt clich altogether. Try a rogue who breaks the law because he believes it to be tyrannical. Or a Paladin who often has trouble with self-control and acts less than noble. It's all about creating a unique but enjoyable persona in your character.
This also counts for major changes your characters will undergo in the game. Epic Destiny in D & D for example is a major milestone. It's the point where the character realizes they are a part of something bigger and begins their transition from mere mortal to near godlike legend. Obviously, things are going to change once this happens. His priorities are going to change, as will his philosophy on life. Try to reflect this in game through his actions or how he interacts with NPCs. Another example would be Shadowrun when a character has to burn an edge point to stay alive. It means that this character was near death, but through some unforeseen circumstance, he survived. Still though, something is changed in him. It is like a part of his died that moment, and he will never be able to get it back. Try to reflect it in role-play.
The role the GM has in role-playing is quite different. The GM needs to make the world come alive for the players in any way he can. For example, when I GM'd my first session of Shadowrun the team was on an assignment to recover an old style audio disk for a shady record label. When the team finally tracked it down, I presented them with a CD-R with several random rock songs burned onto it. It was something small, but it definitely added an extra element of fun to the campaign (in fact, they still have it in their drawer for future reference).
The GM should also be ready to present new quest ideas whenever they might arise. An example would be a friend of mine while running a game of D & D. After the party completed a quest for an extremely powerful dragon, he mentions that the Dragon "bestowed his boon upon the party", boon meaning his blessing or thanks). Not knowing the definition, one of the players casually remarked: "So if we get seven Dragon Boons, do we get a wish?"
The GM thought for a moment, then remarked "You know what? You do now!"
After which, the next several game sessions were spent gathering "Dragon-boons" from six other powerful dragons.
On a final note: it is important to remember that stats should not run a successful role-playing game. They are there as parameters: a measure of a character strengths and limits. Don't get me wrong, they're still important. I've been a part of a few bad sessions where failure to understand game mechanics has made a session confusing and almost unfair. Even so, they have their place. They are needed to keep gameplay fair and balanced, but they should never limit anyone's ability to be creative.
Now what role will you play?