Veteran role-players are familiar with the terms "In Character" (IC) and "Out Of Character" (OOC). The terms In Cavern and Out of Cavern are the Uru equivalents of these terms - they even use the same acronyms. However, they are not exactly the same things.
In traditional role-playing, being In Character means you act as if you are that character - you act as if you know what that character knows. Out of Character is when you step outside that role, and act as yourself as player.
In Cavern is the same sort of thing. But it becomes a bit more complicated with the inclusion of the U are U concept, the idea that you are playing yourself in Uru. Rather than constructing a separate persona, the majority of players in Uru simply 'insert' themselves into the world.
In this context, IC is focused on ones relation to the fiction of the D'niverse as opposed to the persona. Being IC means accepting the D'niverse as real and reacting accordingly. This is an immersive style of play. You are you - you aren't expected to know things you wouldn't normally know or be someone you normally wouldn't be.
U are U is a continuum based on just how true to a player's IC life is to their Cavern life. At one end, the only adjustments are those required for the universe - knowing that D'ni exists, finding a Relto book, etc. As you move towards the other end of that spectrum, more details are adjusted.
One thing to note is that being IC in this manner is separate issue from that of what most people refer to as Story. When people are talking story, they are generally referring to a narrative of some sort - a designed plot with a premise and defined elements (events, characters, themes, etc.). Someone who is IC can be a part of a plot, interact with it and move it along. However, they aren't tied to a plot. They may just be accepting the Cavern as real, immersed in the fiction.
A subset of IC is Role-Playing (RP). Role-playing is a more traditional take on things. Someone has constructed persona that is different from themselves. It is a different 'person' with different skills and motivations than the player playing it. (All of the official characters in Uru fall into this set, except for when Cyantists log in as themselves). Some of these personas are just personalities - people to interact with. But often these characters will be part of a plot.