Pretty well anything that a pilot thinks is putting his flight at risk, and making it unsafe for passengers.

What many people consider to be an emergency landing is oftentimes a precautionary landing. Precautionary landings occur rather frequently and often do not designate an emergency. For instance, warning lights in non-essential systems often warrant a precautionary landing. Unusual engine behavior (thrust or rpm problems), pressurization problems, oil problems, overheating problems, and any one of a whole list may cause the captain to make a precautionary landing. Wing flaps, slats, spoilers, and other secondary flight controls can occaisonally have minor malfunctions that send the plane back to the airport without actually calling for a "mayday" emergency landing situation. With that being said, there are plenty of other problems that do warrant a full-blown emergency. Engine fires, smoke in the cabin, loss of hydraulic control, landing gear failures, loss of wing flap control, etc, will almost always result in the pilot declaring "mayday" and turning around to the nearest suitable runway. The declaration of emergency is something that the captain makes based on the individual situation. If he feels that there is a dangerous problem that puts the aircraft in threat or could cause serious issues later in flight, he will often declare an emergency. In the situation of an emergency or even a precautionary landing, people often get the wrong idea about what is happening. This is partially due to peoples fear of flight but more likely to be due to what the pilot tells the passengers over the intercom. I've heard pilots make ridiculous statements in an attempt to dumb-down information to the point that people are shaking their heads and laughing instead of being comforted.