What is NDE?
Nondestructive evaluation (NDE) is a method to analyze material properties, and locate and characterize flaws within materials, fabricated parts and assemblies. Ultrasonic, radiographic, thermographic, electromagnetic, and optic techniques are used to assess subsurface terrains and internal structures. NDE allows for monitoring and imaging internal/ external conditions of mechanisms and dynamic experiments.

In Layman's terms, sniff, tap, examine an apple in the grocery store, you're conducting nondestructive evaluation!

Other terms:

NDT - nondestructive testing

NDI - nondestructive inspection

Who uses NDE?
The following are just a few industries using NDE technologies.
Aerospace
Nuclear Power Plants
Automotive
Pulp and Paper Mills
Defense
Refineries and Chemical Plants
Fossil Power Plants
Ship Building

Nondestructive vs. Destructive
When is it appropriate to use "destructive" evaluation vs. "nondestructive"? Destructive testing is
a valuable part of being able to qualify an initial sample evaluation to develop a nondestructive method.

Back to the apple example:
Destructive
To determine the ripeness of an apple, you cut open for examination. What are the deciding factors?
Smell, color and firmness of the flesh of the apple. Then you either eat the apple or throw it away.

Nondestructive
To determine the ripeness of an apple, you sniff it for ripeness, grip it for firmness, examine the exterior
for bruises. Now the apple may be eaten or left to ripen.

Destructive testing is valuable; nondestructive evaluation is invaluable.

Nondestructive evaluation...
Provides a preventive measure against product liability
Minizmizes scrap loss
Improves product performance