In the beginners tutorial, you've seen the Arcade physics system in action (only a tiny little bit, though). Phaser however takes physics to a whole new level, and brings 2 different systems with it: Arcade, p2.
Arcade: the arcade physics system is the easiest of them all (which also means the most limited), and as you might have guessed: best suitable for arcade games. It has simple collision and overlap checks for rectangle bodies (called AABB), and features like giving bodies velocity, gravity, friction, bounce, etc.
Learn all about it in the Arcade Physics Tutorial
p2: the p2 physics system is the most advanced system. It does everything arcade does too, but can do a whole lot extra. It's very realistic yet easy to implement. The p2 system is well suited for the complex physics in a game (realistic falling/bouncing of objects, chains, thrusts, springs, constraints, etc.).
Learn all about it in the p2 Physics Tutorial