While I do agree with the previous answers, which I will not repeat
here, I feel there is a dimension that is missing, or at least left
implicit. I think it should be explicited.
You could have the problem of finding the product of 222 by 333 for
some purpose. In that case, all you need is an answer (73926). Why
bother with complicated explanation as to why that is the product? (assuming you can trust your source for the answer)
In general, an algorithm addresses a family of problem, such as:
"given two integers $x$ and $y$ (with such and such representation),
how do I compute (a representation of) the product $x\times y$. The
algorithm can then be used to solve all the problems in the family,
one of them being the product of 222 by 333.
The interest of algorithms actually stems from the fact that they
solve a whole family of problems. This is remindful of the chinese
proverb "Give a man a fish he will eat for a day. Teach a man to fish
he will eat for a lifetime".
The fact that algorithms address families of problems is the source of
many of the concepts of the study of algorithms such as decidability
(is there an algorithm that solves all the problems in the family) or
complexity (what is the cost in time, memory, or some other relevant
unit) of applying the algorithm, given some measure of the size of the
input, i.e., of the parameters of the family.