Regular characters are those that appear in every episode - or the majority of episodes - of a series. The actors that play regulars are usually contracted into every episode of a series, although they may drop out of an episode from time to time, either because of outside forces or because the writers simply can't think of anything to do with them. (When this doesn't happen and the actor makes a perfunctory appearance anyway, the Mandatory Line is usually the result.)
Usually, though, even if a Regular Character doesn't have a big role in the story, they will get at least a few lines or background appearances to comply with their contract. Sometimes a regular character will be shifted to a sister show on the same network (for example, Spike and Oz crossing over from Buffy to Angel), allowing them to fill out the contracted number of episodes even if they're not in every episode of that one series.
However, dropping them entirely from episodes is currently more common than having token appearances, with shows like Lost and Heroes only having a handful of episodes each season-if any-featuring all regulars. However, this is down to those shows have starring casts that at their peak can reach 16 people or more. The number dropped can be extreme: The Heroes episode "Company Man" features only three series regulars, for example.
Sometimes Recurring Characters - those that don't appear in every episode of a series - will become Regular Characters. And sometimes it works the other way, if a regular leaves a series and returns in a supporting role, or if the actor playing a regular gets another job and can't commit to a permanent regular role.