Elongation to failure is a measure of the ductility of a materials, in other words it is the amount of strain it can experience before failure in tensile testing.
A ductile material (most metals and polymers) will record a high elongation. Brittle materials like ceramics tend to show very low elongation because they do not plastically deform.
Rubber extends by a large amount before failure, but this extension is mostly elastic and is recovered.
Design issues
Elongation is important in components which absorb energy be deforming plastically (e.g. crash barriers, car bumpers).
High elongation to failure is important for "plastic hinges" (e.g. video cassette boxes).
Elongation is important in manufacturing - it measures how much bending and shaping a material can withstand without breaking.
Measurement
Tensile testing is used to find many important material properties. The compression test is similar but uses a stocky specimen to prevent bending.
Units & Values
Because elongation is equal to the failure strain it has no units, but is often given in % strain.