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The term observer has a number of non-equivalent uses in science.
Special relativity 
The term observer in special relativity refers most commonly to an inertial reference frame. In such cases an inertial reference frame may be called an "inertial observer" to avoid ambiguity. Note that these uses differ significantly from the ordinary English meaning of "observer". Reference frames are inherently nonlocal constructs, covering all of space and time or a nontrivial part of it; thus it does not make sense to speak of an observer (in the special relativistic sense) having a location. Also, an inertial observer cannot accelerate at a later time, nor can an accelerating observer stop accelerating.
General relativity 
In general relativity the term "observer" refers more commonly to a person (or a machine) making passive local measurements, a usage much closer to the ordinary English meaning of the word.
Quantum mechanics 
In quantum mechanics, "observation" is synonymous with quantum measurement and "observer" with a measurement apparatus and observable with what can be measured. See also quantum mind-body problem, measurement in quantum mechanics and Schrödinger's cat.
Thermodynamics and information theory 
See for example Maxwell's demon.