My boring professor said, "A demurrer is a motion to dismiss for failure to state a claim."
My more interesting professor said, "A demurrer is a lawyer's way of saying 'so what?'"
To put it in more concrete terms, a demurrer is a motion requesting that a claim be dismissed from court, on the basis that the alleged wrongdoing isn't against any law. A demurrer doesn't contest the facts being presented: it merely says that "even if all of this is true, you can't sue me for it."
For instance, let's say you're driving down the street. Someone walking next to your car trips over their shoelaces and falls on their face. They happen to know a (particularly unskilled) lawyer, and they sue you to get compensation for their injury. Since you can't be sued for something that happens merely in the vicinity of your car, which you had no role in causing, you can "demur"—that is, formally ask the judge to dismiss the claim.
As you might infer from the example, demurrers are often used to defend against clearly frivolous lawsuits. However, there are many cases that might seem to be completely substantive, but that still get demurred out of court. Often, this happens because common law (case law) can be interpreted differently by different people: the plaintiff might think they have a case for negligence or breach of contract when the defendant's activity wasn't technically negligent or contractual to begin with, based on the long and obfuscated history of rulings in that jurisdiction.
The term is an old one, dating back to medieval Britain (it comes from the Latin de morari, "to delay indefinitely"), and in many jurisdictions (including U.S. federal courts, but not some state courts) its use is now deprecated in favor of the more general motion to dismiss.