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There is a degree of instability in any measurement, so that if it is repeated, it is possible that a different result may be obtained. Such instability, generally described as 'measurement error', may affect the conclusions drawn from an investigation, and methods exist for allowing it. It is less widely known that different disciplines, and different workers within a discipline, have different interpretations of the term 'measurement error', and that these different interpretations are liable to give rise to different results. This article is an attempt to describe and systematise a number of conceptualisations of measurement error, and the associated concept of randomness, and to show how these work out in practice. Because of the importance of this topic, the aim has been to make the descriptions accessible, avoiding jargon and other technical matters such as equations as far as is feasible.