Not Very Effective refers to a Move that deals less than normal damage to a Pokémon.
When a Move is used on a Pokémon in battle, it may deal normal damage, less damage (Not Very Effective) or increased damage (Super Effective). This is determined by the Type of Move used and the Type of Pokémon the Move is used on, also known as Type Effectiveness.
Certain Pokémon Types are more resistant to certain Types of damage. For example, Charmander is in battle against a Squirtle. It decides to use the Move Ember, which happens to be a Fire Type Move. Being that Squirtle is a Water Type Pokémon it is strong against Fire and therefore it will take reduced damage from the attack.
If a Pokémon Type is resistant to a Move, "Not Very Effective" will appear on the screen after the attack lands.