Both characters and dragons have various statistics that affect their performance in battle (although dragon's stats are far more complicated). Here goes.
LV: Experience level; this goes up as you gain EXP in battle. For characters, their other stats are directly dependent on this one and their equipped items. For dragons, the only stats that raise when they gain levels are HP, MP, and one elemental (depending on the dragon). For EXP information, see the bottom of this page.
HP: Health Points. When they reach 0 the unit is dead. When all characters in a unit are dead, the unit disappears from battle for the rest of the battle. Naturally if a dragon's HP drops to 0 it also disappears, meaning the associated character party loses more or less all their skills, techniques, and special effects. You can raise a dragon's max HP by feeding it drugs or any item that sounds like a foodstuff, but mushrooms give you the best bang for your buck.
MP/SP: These two stats are identical; they're used to cast spells or use techniques. Note that there is NO WAY of refilling a dragon's MP in battle (the only possible way is if the Mini-Devils use Complete Heal All in their Dance). You can raise a dragon's max MP mainly by feeding it Magic Gins.
Attack, Defense, Speed, Magic: Self-explanatory. For dragons, these are contingent on their Strength, Vitality, Dexterity, and Mind stats, respectively.
Each Elemental gains one level every 10 points, to a maximum of 100 (level 10). You can raise elementals by feeding it its particular Grass or any equipment that's that elemental (e.g. Ice Brand for Ice, Diamond Armor for Thunder, etc.). Thunder, Life, and Poison can be easily raised by feeding it mushrooms, too.
Fire, Ice/Water, Thunder: These are the three "normal" elementals. They affect a dragon's strength of their elemental spell, and the strength of any elemental techniques of the associated party in battle. Note that in this shrine, Ice and Water are used interchangably.
Life: Also known as Recovery. This affects the strength of the dragon's/characters's healing or drain-HP techniques. It also affects whether the dragon will be in Holy or Dark form (see Dragons).
Poison: Also known as Death. (Note, however, that for enemies Poison and Death are two different attributes!) This affects poison skills (any poison skill also has a chance of poisoning the enemy). This also affects Holy vs. Dark forms.
Earth: This stat is invisible. It's equal to the lowest of the three levels of Fire, Ice/Water, and Thunder.
Holy: Another invisible one. It's always either level 0 (for most dragon forms) or level 10 (for Holy and Master forms).
Dark: Again invisible. It's either 0 (for most forms) or level 10 (for Dark and Master forms).
Transform: This is kind of a pseudo-elemental. The Transform stat (invisible) can only be raised using the "Princess's ???" item (which raises it by 20), and can only be lowered using the "Gunso's ???" item (which lowers it back to 0 no matter what it is). There are five Princess's ??? items in the game, so using all five will raise Transform to level 10 (at 100 points, similar to other elementals). This means that Transform can only be at level 0, 2, 4, 6, 8, or 10. A dragon whose Transform stat is higher than 0 is automatically an "Unknown Form" dragon (see Common Dragon Forms). Also, any character whose dragon is an Unknown Form has all his techniques/spells replaced with a single non-elemental technique/spell called "Unknown (something)". This raises levels with the Transform stat as well. When Transform is level 10, the dragon turns into a Behemoth, which is just the highest-level Unknown Form.
Each of these stats gains a level every 25 points, to a max of 250.
Strength: This affects a dragon's physical strength in battle, as well as its Bite skill. Use weapons to raise this.
Vitality: This affects its defense in battle, as well as its Body Blow skill. Use armor to raise this.
Dexterity: This affects how quickly it acts in battle as well as its accuracy and its Claw skill. Use rapiers or mushrooms to raise this.
Mind: This affects how strong its spells are, as well as the Sonic Boom skill. Use books, mushrooms, rods, or robes to raise this.
Each of these stats gain a level every 10 points, to a max of 100. However, these stats can also be lowered by some items, whereas the other stats above cannot (other than Transform)
Fidelity: When this reaches level 10, the dragon becomes a Uni Uni (see Common Dragon Forms). I'm unsure if it has any other effects in battle... but I *think* Fidelity may affect how often a dragon appears before you to hurt the enemy when you're in a melee battle. Use a Heal to return a dragon to his normal form.
Loyalty: If a dragon is a Uni Uni, this is basically the "elemental level" of the dragon. In other words, the dragon's Attack and Uni Breath will be the Loyalty level, as will all the skills and techniques of the associated character party. If Loyalty is level 10, the dragon will turn into a Black Dragon and gain an extra spell in melee battle. I'm unsure if Loyalty affects anything else in battle...but I *think* this may affect what a dragon will do when he appears in a melee battle (i.e. if it's low he may just "watch" or "encourage" instead of attacking).
Wisdom: Also known as Intelligence. This affects how smart a dragon is. A dragon with high wisdom will use its stronger spells and it'll use spells the enemy is weak to. Books will raise this.
Intimacy: Affects how much the dragon likes you. If it's too high the dragon will stay too close to you (and it makes trying to get your party to move together pretty tough). If it's too low the dragon will stay further away from you, and I think it'll also refuse to heal you or some such. If it drops to 0, it'll become a Lonely Dragon. "Sweet" items will raise it, "bitter" items will lower it. Intimacy may also drop if you continually feed other dragons while ignoring one dragon.
Personality: This is completely invisible; there's no way to check it. Basically, if it's high dragons will enter melee battles more often, and if it's low they'll use field attacks more often. Feeding them weapons will raise this, and feeding them armor will lower it.
Characters and dragons will gain EXP based on their actions in battle. Note that an action does not have to be successful to gain EXP! EXP is calculated as follows.
(Level x Level x Multiplier) + Enemy's EXP Amount
By "Level", I mean the experience level of the character or dragon, not the level of the technique. The Enemy EXP Amount varies per enemy (see Enemies, and is only added if you *defeat* an enemy, not if you just hit it). The Multiplier depends on the action.
That's for in the field. Note that the entire party gets the same amount of EXP regardless of how many characters execute the technique. (Ice Magic will give the party the same amount of EXP regardless of how many Wizards you have in your party.) But note that if you have more than one party member executing the technique, the "Level" used above is the average of their experience levels, rounded down.
Now for in melee battles with characters. Each character now has their own EXP count. However, any enemies defeated will have their EXP added to every character. Also, if a character Defends, each time an enemy attacks him he will gain (Level x Level) EXP. However, if a character Defends and is not hit, he won't get any EXP at all for that battle (unless enemies are killed by other characters). Similarly, if you select an attack but all enemies die before the attack can be done, you won't get EXP for doing the attack. Finally, if a dragon attacks before battle, the characters will gain the EXP of the spell/attack, not the dragon.
For more specific info on multipliers for techniques and spells, see Techniques and Spells.