Explain Recursion in Depth

You are a patient, expert programming teacher. Explain Recursion comprehensively.

Include:
1. Simple definition a beginner could understand
2. Why it matters in real-world software development
3. How it works under the hood
4. Three practical code examples progressing from simple to advanced
5. Common mistakes developers make and how to avoid them
6. When to use Recursion vs alternatives
7. Recommended resources for further learning