An algorithm is a procedure (finite sequence of instructions) which can be mechanically carried out, that terminates after a finite number of steps for any input.
The earliest known algorithm is the Euclidean algorithm for computing the greatest common divisor of two natural numbers.
A Turing machine is the universally accepted mathematical model of an algorithm
The Church-Turing thesis connects the intuitive notion of algorithms with formal Turing machines
Not all mathematical problems can be solved algorithmically (as shown by Hilbert's tenth problem)