By defeating enemies your party gets experience points that are needed to increase your level. The amount earned by each character depends on their level. If the character's level is low, they'll receive more points:
Leveling up increases a character's stats and they may also learn new skills or traits.
The basic stats that increase on level up are:
HP (Health Points): if it reaches zero, the character will be out of battle.
MP (Mana Points): needed to cast magic.
Strength: affects your physical (weapon) damage.
Defense: reduces physical damage received.
Spirit: increases the damage or healing of magic spells.
Resistance: reduces magical damage received.
Agility: affects the charging speed of your turn.
Initiative: defines how much TP (Tech Points, necessary to use non-magical skills) you will start the battle with.
Other three stats -that don't increase by levelling up- can be affected by traits, skills and equipment:
Accuracy: the precision of your physical and magical attacks.
Critical: the chance to inflict double damage to an enemy.
Evasion: the chance to evade both physical and magical attacks.