Newton’s method is an example of an algorithm: it is a mechanical process for solving a category of problems (in this case, computing square roots).
To understand what an algorithm is, it might help to start with something that is not an algorithm. When you learned to multiply single-digit numbers, you probably memorized the multiplication table. In effect, you memorized 100 specific solutions. That kind of knowledge is not algorithmic.
But if you were “lazy”, you might have learned a few tricks. For example, to find the product of \(n\) and 9, you can write \(n-1\) as the first digit and \(10-n\) as the second digit. This trick is a general solution for multiplying any single-digit number by 9. That’s an algorithm!
Similarly, the techniques you learned for addition with carrying, subtraction with borrowing, and long division are all algorithms. One of the characteristics of algorithms is that they do not require any intelligence to carry out. They are mechanical processes where each step follows from the last according to a simple set of rules.
Executing algorithms is boring, but designing them is interesting, intellectually challenging, and a central part of computer science.
Some of the things that people do naturally, without difficulty or conscious thought, are the hardest to express algorithmically. Understanding natural language is a good example. We all do it, but so far no one has been able to explain how we do it, at least not in the form of an algorithm.