Anaerobic Threshold Running
Anaerobic or Lactate Threshold running is normally the domain of middle and long distance runners.
Here, the session is meant to be run as fast as possible, whilst remaining entirely aerobic (in other words, the body is replenished with as much oxygen as is being used during the activity - at no stage will the body go into oxygen debt). To accomplish this is often difficult, as it is not easy to tell whether you are running entirely aerobically.
The use of a heart rate monitor can be helpful, although the actual limit will vary considerably from person to person - an averagely fit person may find this limit is 140-150 beats per minute, whilst a top athlete may be able to raise their heart rate up to 180.
A simpler way to tell whether you are running aerobically - which is relatively accurate - is that you should be able to carry on a relatively normal conversation. If you are too out of breath to talk properly then you should slow a little.
The thing to remember with threshold runs is that you are trying to run hard without being limited by the amount of oxygen you take in. These runs will be important for any distance runner to build fitness, and should be hard work, but you should not be running so hard that you feel that you need to stop because you are completely out of breath. Another way of thinking about it is that you should run hard, but know that if you needed to produce a sprint finish you could.
Many athletes refer to these types of sessions as "Tempo" runs.