I've never heard the term "finiteness" applied to an algorithm, but I think that's because the definition of an algorithm includes that it must be finite. So think of any algorithm and there is your example of finiteness.
An algorithm is just about any method used to solve a problem. Common algorithms include the methods taught in school to add, subtract, multiply and divide numbers with pencil and paper.
Euclid's algorithm is an easy way to find the greatest common divisor (GCD) of two numbers. Let's try 1029 and 375. If I have two numbers a and b, where b is smaller than a, the process looks like...
An algorithm is a stepwise sequence of operations that can be performed to solve a problem. The operations are expected to be 'simple', so they can be performed mechanically.
A simple example...