NSTEMI is an acronym meaning "non-ST segment elevation myocardial infarction," which is a type of heart attack. This is determined by a electrocardiogram (ECG) test.
Myocardial infarctions (heart attacks) occur when a coronary artery suddenly becomes occluded by a blood clot, causing at least some of the heart muscle being supplied by that artery to become infarcted (that is, to die). Myocardial infarctions are divided into two types, according to their severity. A NSTEMI is the less severe type.
In a NSTEMI, the blood clot only partly occludes the artery, and as a result only a portion of the heart muscle being supplied by the affected artery dies.
In contrast to the more severe form of heart attack (the STEMI), the NSTEMI does not produce characteristic elevation in the "ST segment" portion of the ECG. (ST segment elevation indicates that a relatively large amount of heart muscle damage is occurring, because the coronary artery is totally blocked). This means that in a NSTEMI, the artery is only partially blocked.
A common problem when a patient has an acute coronary syndrome without ST segment elevation is deciding whether an actual heart attack is occurring or instead whether the patient is simply having unstable angina. Measuring cardiac enzymes, which reflect heart muscle damage, is an important tool in making this distinction.
You can read about the more severe form of heart attack - the STEMI - here.