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What is an algorithm?

Informally, an algorithm is any well-defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values, as output. An algorithm is thus a sequence of computational steps that transform the input into the output.

An algorithm should have three important characteristics to be considered valid:

It should be finite: If your algorithm never ends trying to solve the problem it was designed to solve then it is useless
It should have well defined instructions: Each step of the algorithm has to be precisely defined; the instructions should be unambiguously specified for each case.
It should be effective: The algorithm should solve the problem it was designed to solve. And it should be possible to demonstrate that the algorithm converges with just a paper and pencil.