Definition of an Algorithm

An algorithm is a well-ordered collection of unambiguous and effectively computable operations that when executed produces a result and halts in a finite amount of time.

Well-ordered means the steps are in a clear order.
Unambiguous means the the operations described are understood by a computing agent without further simplification.
A computing agent is the thing that is supposed to carry out the algorithm.
Effectively computable means the computing agent can actually carry out the operation.

This definition comes from An Invitation to Computer Science (Gersting/Schneider)

If we can specify an algorithm to solve a problem then we can automate its solution. That which carries out the steps of an algorithm (a computing agent) just needs to be able to follow directions; not understand the concepts or ideas behind the algorithm.

The word algorithm is derived from the last name of Muhammad Al-Khowarizmi (full name: Abu Ja'far Muhammad ibn-Musa Al-Khowarizmi), a famous Persian mathematician and author of the eighth and ninth centuries. (His most famous book is Kitab al jabr w'al muqabala.)