"We may at once admit that any inference from the particular to the general must be attended with some degree of uncertainty, but this is not the same as to admit that such inference cannot be absolutely rigorous, for the nature and degree of the uncertainty may itself be capable of rigorous expression."
— R. A. Fisher (1966, 8th ed., p. 4) The Design of Experiments
Introduction
Measurement is an experimental process that produces a value that can reasonably be attributed to a quantitative property of a phenomenon, body, or substance.
This property that is the object of measurement (measurand) has a numerical magnitude and a reference that gives meaning to that numerical magnitude: for example, the mass of the International Prototype Kilogram is 1 kg.
That process involves direct or indirect comparison with a standard (for example, a certified 32768 Hz quartz crystal resonator, or a primary gas mixture with 1.079 micromole of methane per mole of air).
This comparison typically is accomplished by making the phenomenon, body, or substance of interest interact with a measuring instrument capable of producing an indication that is responsive to the property of interest. If the instrument has been calibrated, then the indications that it produces are meaningful in relation with a relevant standard.
The GUM defines measurement uncertainty as a "parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand''. The VIM defines it as a "nonnegative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used''.
For scalar measurands (that is, when the property of interest can be quantified by a single real number), the VIM suggests that this parameter may be, for example, a standard deviation called standard measurement uncertainty (or a specified multiple of it), or the halfwidth of an interval that includes the measurand with a stated coverage probability. (The expression standard measurement uncertainty is reserved for measurement uncertainty expressed as a standard deviation.)
This suggestion follows from the position that measurement uncertainty expresses incomplete knowledge about the measurand, and that a probability distribution over the set of possible values for the measurand is used to represent the corresponding state of knowledge about it: in these circumstances, the standard deviation aforementioned is an attribute of this probability distribution that represents its scatter over the range of possible values.
This position is expressed in the GUM (3.3.1), where it is suggested that measurement uncertainty "reflects the lack of exact knowledge of the value of the measurand''. The corresponding state of knowledge is best described by means of a probability distribution over the set of possible values for the measurand.
This probability distribution incorporates all the information available about the measurand, and indeed expresses how well one believes one knows the measurand's true value (a felicitous interpretation due to Dr. Charles Ehrlich, from NIST), and fully characterizes how the degree of this belief varies over that set of possible values.
It is this distribution that imparts meaning to the parameter that is chosen to quantify measurement uncertainty. For example, the CODATA 2006 estimate of the value of the StefanBoltzmann constant is σ = 5.670400 x 10^{8} W m^{2} K^{4}, with corresponding standard measurement uncertainty u(σ) = 0.000040 x 10^{8} W m^{2} K^{4}. If the probability distribution that represents the state of knowledge about this constant should be Gaussian (or, normal) with σ as mean and u(σ) as standard deviation, then this would suggest that the probability is 68% that the constant's true value lies within u(σ) of that estimate.
When the measurand is a vector, rather than a scalar quantity, or when it is a quantity of even greater complexity (for example, a function, as in a transmittance spectrum of an optical filter), then the parameter that expresses measurement uncertainty will be a suitable generalization or analog of the standard deviation.
Tutorial for Metrologists on the probabilistic and statistical apparatus underlying the GUM and related documents 
Uncertainty Machine 

The following link to the browser version uses MathML. Mozilla Firefox has builtin capabilities to render it, but Microsoft's Internet Explorer (IE) does not, unless the MathPlayer plugin is loaded. Both Mozilla Firefox and Microsoft's Internet Explorer require that the STIX fonts be installed. Tutorial for Metrologists (Firefox/IE version) Tutorial for Metrologists (Adobe PDF version) 

References
 Barry N. Taylor and Chris E. Kuyatt (1994). Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. NIST Technical Note 1297.
 Evaluation of measurement data – Guide to the expression of uncertainty in measurement JCGM 100:2008 (GUM 1995 with minor corrections)
 Evaluation of measurement data – Supplement 1 to the "Guide to the expression of uncertainty in measurement" – Propagation of distributions using a Monte Carlo method JCGM 101:2008
 M. R. Bottaccini (1975). Instruments and Measurement. Charles E. Merrill Publishing Company, Columbus, Ohio.