What is recursion and when is it useful?

Recursion is the repeated application of a process. In JavaScript, recursion involves functions that call themselves repeatedly until they reach a base case. The base case breaks out of the recursion loop, thus allowing previous calls to the function to return a result. If no such case exists, the function will call itself indefinitely resulting in a stack overflow.
Recursion is used to solve problems where the solution depends on solutions to smaller instances of the same problem. A commonly-used example of a problem that can be solved recursively is the Fibonacci sequence:
The base case for this example is n being less than or equal to 1, where the function returns the value of n. Any other value will call the fibonacci function twice to compute the values of n - 1 and n - 2. These will in turn call the fibonacci function again until the base case is reached.
@Further reading
While the Fibonacci sequence can be solved with recursion, it can be more efficient to solve it using iteration. However, the inverse is true for a lot of other problems where identifying and indexing the sub-problems is difficult or costly.

const fibonacci = n => {
  if (n <= 1) return n;
  return fibonacci(n - 1) + fibonacci(n - 2);
};

fibonacci(6); // 8

The base case for this example is n being less than or equal to 1, where the function returns the value of n. Any other value will call the fibonacci function twice to compute the values of n - 1 and n - 2. These will in turn call the fibonacci function again until the base case is reached.
@Further reading
While the Fibonacci sequence can be solved with recursion, it can be more efficient to solve it using iteration. However, the inverse is true for a lot of other problems where identifying and indexing the sub-problems is difficult or costly.

From 30-seconds-of-code: recursion