Jedi Realism is the concept of the more philosophical side of being a Jedi - in other words the Jedi mindset - as opposed to purely role-playing as a Jedi. To someone who believes or follows this path, it is more important that his or her actions are based on the teachings of the Jedi than simulating the look and talk of a Jedi. In other words it is more like being rather than seeming.
Jedi Realists often take their discussions and research from various real life sources rather than relying on just the philosophies described by the Jedi in Canon. These may include Buddhism, Hinduism, martial arts, ethical/philosophical organisations and various other sources.
SL Jedi Realism vs Role-play?
To an extent, most Jedi Orders in SWRP have an element of Jedi Realism; however it is certainly true that there are different levels of emphasis on this, especially when placed against the context of role-playing. In extreme cases, some Jedi in SWRP and some Realists consider Realism to be completely against role-playing; others on the far side of the spectrum consider it perfectly reasonable for a Jedi to be just a role to play and that the rest of what someone does in SL and RL is irrelevant. Most Jedi, however, would be between these two positions, feeling that there is no inherent conflict between being a realist and being a role-player (but varying on which way they lean).
The Jedi Order in SWRP perhaps best known for this last position and which tries to ensure a Realist base for everything they do including role-playing is Jedi Guardians of the Peace, based in the Ossus sim.