If the input to a linear system has LaplaceTransform X(s) and the LaplaceTransform of the output is Y(s), then the TransferFunction H(s) is defined as

  H(s) = Y(s)/X(s)

where s is the complex-frequency variable

Pictorially it can be represented as:

  X(s) --[H(s)]--> Y(s)

System here is in the general sense, such as Mechanical, Electrical, Thermal, Fluid, Economic, Social, ''et cetera''. Networks of components can be assembled such that the overall TransferFunction can be derived from the TransferFunction of individual parts (then the resulting system can be treated as a BlackBox) but the whole is not necessarily the sum of its parts.  The result can be EmergentSystems.
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The key thing here is that the underlying system is linear. A non-linear system can be linearized around a base point, but the linearization is an approximation which will only be valid in the region of the base point.

* No. At least not in general. The transfer functions can model most sub-exponential impulse responses not only linear ones. The main limitation usually is that they are 0 for t less some t0. 
** An example that can be fully captured by a TransferFunction:
http://upload.wikimedia.org/wikipedia/de/5/5a/Standartregelkreis.png
-- GunnarZarncke (has had ControlEngineering and knows Laplace and Z transforms)

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See also NyquistStabilityCriterion
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CategoryMath