Details:
As you play through the Ferrum League and Single Battle, you will discover that your Pokémon is earning experience. This will level it up with the ability to get all the way up to Level 100. As you level your Pokémon up, you will receive titles, as well as a medal for the character selection screen. However, the most notable thing about Skill Levels are the Skill Points. Skill Points basically allow you to increase your Pokémon's capabilities in one of each of the following stats

• Attack: Boosts your attack and the damage dealt
• Defense: Boosts your defense and reduces damage you receive
• Synergy: Increases duration of Synergy Burst and increases the boost that both Attack & Defense get during Synergy Burst
• Strategy: Increases the speed Support Gauge fills and boosts the damage inflicted by Support Pokémon.

You can, however, alter the Skill Level points you have assigned Pokémon within the My Town option of the game. This will allow you to reorganise each of the points and saves it.

The following modes allow for you to use the Skill Levels in your battles

• Ferrum League - Non Optional
• Special Events - Non Optional
• Single Battle - Optional
• Local Battle - Optional
• LAN Battle - Optional

It's currently being tested as to whether Skill Levels are implemented online. Practice Battles have it forced off

In addition to just grinding the Ferrum League and Single Battles to increase your Pokémon's Skill Levels, there are now other ways. Daily Challenges are special once per day challenges that allow for you to use varied Pokémon. Upon completion of the challenge, you will often receive 1 or 2 Skill Level increases for the Pokémon you used, as well as a random other Pokémon. This allows for you to level up your Pokémon far quicker than before.