Can anybody give me a good definition of "mathematical algorithm" and some examples of it?

An algorithm is like a recipe. It tells you how to do something without any depth or understanding of why it works. People can memorize and apply algorithms, but will usually forget them after a short time. On the other hand, people who understand the reasons why the algorithms work will usually remember it. That's why, for example, some people never forget how to do fractions, and others forget as soon as they take the test.