Hi there,
There are several ways to roleplay in an MMO, let's look at these not any other ways there might be
RPing in MMOs can be seen as most close to Pen & Paper roleplaying.
In Pen and Paper you often have a Game or Dungeon Master telling a story in which a character created based on some rules is played by you as a player. Famous P&P systems are D&D (Dungeons and Dragons) or Shadowrun. Most of them are played with help of Dice, but there are also other systems, e.g. in "Engel" there is an Aracana System in which you determine the outcome of events with help of Tarot-like cards.
In MMOs you do most likely not need extra tools to determine what will happen exactly.
"Live" RPing
By this I mean RPing lvie in a group with help of text or voice chat.
One of the most important points when it comes to "live" RPing is that you don't decide what happens.
E.g. you can say "I'll attack him with the intent to kill him." but you can't say "I kill him", that is decide by the dice (or, in RoM, by the use of your skills in combat).
Action are often put in *...* e.g.
"Ah yes *sits down* this is the best time after battle: Ale time! *gives a deep coughing laugh*"
"Story" RPing
Giving your character a background story is also often called RPing. This is most of the time a short fantasy story featuring the early years of the character a person is playing or his adventures. Eg. you could write about fighting in Kulech:
"I'm shivering. My heart is beating fast. We came here to fight our way through these abnormal creatures to gain honor and find treasures, but all I'm hoping for right now is survival. Our healer is down. Her dress is thick with blood. Her leg lies there in a weird ancle. She's hardly breathing. There comes another one. Oh Ayvenas, help!"
RPing can be a lot of fun if all participants agree on the same rules.
I hope I could add some info you find helpful
~Laghmara