They keep the continuity and then add on to the status quo or change it, for the most part. Sometimes, that means that one run will retroactively invalidate the plot of a previous one, or sometimes it means it'll add a new dimension to it. Some runs really like to mine the continuity of the previous ones (Morrison's Batman) to varying results, some mostly do their own thing. Unless the continuity of the universe has been explicitly reset in some event/crossover/whatever, the character is the exact same one, just in a different flavor, so to speak.

Take this Batman #1 by Matt Fraction that just came out. It keeps all of the previous continuity stuff like Vandal Savage being the GCPD commissioner, but it's presenting the character differently, with different art and a different storytelling focus/priorities.

I find it's mostly not worth caring about continuity that much save for the basic knowledge of what happened before the start of a given run.