Algorithm
The word "algorithm" itself derives from the 9th century mathematician Muḥammad ibn Mūsā al-Khwārizmī, Latinized "Algoritmi". A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem (decision problem) posed by David Hilbert in 1928. Later formalizations were framed as attempts to define "effective calculability" or "effective method". Those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, and Alan Turing's Turing machines of 1936–37 and 1939.
Algorithm characterizations
A careful reading of Gurevich 2000 leads one to conclude (infer?) that he believes that "an algorithm" is actually "a Turing machine" or "a pointer machine" doing a computation. An "algorithm" is not just the symbol-table that guides the behavior of the machine, nor is it just one instance of a machine doing a computation given a particular set of input parameters, nor is it a suitably programmed machine with the power off; rather "an algorithm is the machine actually doing any computation of which it is capable". Gurevich does not come right out and say this, so as worded above this conclusion (inference?) is certainly open to debate: