The level is the main way in which the game is presented through the single-player experience. Each player starts the game from Level 1. The level is indicated on the left side of the main menu of the game inside a star-shaped object. A new user will see the following status:
The 70 written there means that the player needs 70 more points in order to proceed to Level 2.
The figure below represents a more advanced user:
This user is already at Level 2 and he has in his possession some Points (the bar has progressed a bit) and needs another 344 more Points in order to proceed to Level 3. Every time you advance a Level you can see the Points needed to unlock the next one. As you level up, you unlock new Game Modes and other services (e.g. tournaments, bigger bonuses, free Diamonds, etc). But most importantly, you gain prestige within the community of the players of the game!
Whenever you play against an opponent, you can see his level. It is the number inside the star-shaped object shown in his profile.