As part of Pokemon's "something for everyone" appeal, you have your monsters that are supposed to be cute like Pikachu as well as ones that are supposed to look super intimidating and badass. Underneath all of that, a Pokemon's usefulness in battle is based on these factors, which I'll try to quickly bring you up to speed on using this example:
1.
Type. A Pokemon can have 1 or 2 out of the 18 possible types, and all attacks have a type as well. This operates on a Rock-Paper-Scissors type of system where all attacks have a x0, x0.5, x1, or x2 damage multiplier against a certain type. When a Pokemon has two types, an attack's effectiveness is multiplied against both of those types. For example, Fire-type attacks are super effective (x2 multiplier) against Ice-type Pokemon and not very effective (x0.5 multiplier) against Water-type Pokemon, so a Fire-type move against a Water/Ice type Pokemon would do normal (x1) damage. Additionally, attacks that are the same type as the user receive a 50% damage bonus.
2.
Stats. Each species of Pokemon has a base value for each of the stats listed (HP, Attack, Defense, Special Attack, Special Defense, and Speed). While this gives a general footprint, there's a lot of variation introduced through individual values (the Pokemon equivalent of genes, all of these are optimized for both my Pokemon and the opponents' in the Maison, so I won't go into too much detail there), natures (which raise one stat and reduce another by 10%), and effort values (essentially a limited amount of bonus stat points you can spend as you please).
As an example of the variation both within and among individuals, the Glalie in the screenshot above has the same base stat (80) for Attack, Defense, Special Attack, Special Defense, and Speed. As you'll see later, these stats can be modified (from a 0.25x to a 4x multiplier) by certain battle conditions. Additionally, Pokemon have Accuracy and Evasion multipliers, which all start out at 1x, that affect the likelihood of an attack connecting.
As far as how these stats are used in battle, HP is the old RPG standby; if you run out, you're dead. In addition to having a type, attacking moves can be either physical (the straightforward punching, hitting type of moves) or special (equivalent to elemental/magic attacks in other RPGs). Attack/Defense is a measure of a Pokemon's ability to dish out or take physical attacks, and likewise for the special counterparts. Speed is perhaps the most important stat for a turn-based RPG, determining which Pokemon gets to move first; if both Pokemon have the same Speed, the turn order is a coinflip.
3.
Ability. These are passive traits that can take on any number of different effects. The best abilities can make Pokemon with mediocre stats among the most fearsome characters in the game, and the worst ones can completely neuter Pokemon with top-tier stats and attacks. Pokemon can only have one ability at a time, but some species of Pokemon can potentially have one of up to three different abilities which may significantly affect the optimal strategy to use against them.
4.
Item. Each Pokemon is allowed to hold an item, which can confer offensive or defensive benefits. Item choice is extra important because using any of the potions or healing items that are prevalent during the story mode is prohibited in the Battle Maison, and no two Pokemon on your team may hold the same item.
After these four, each Pokemon can use up to four attacks. The aforementioned physical and special attacks can vary by their base power, accuracy, Power Points (the number of times in a battle they can be used), and side effects. They can also land critical hits, which deal 50% more damage while also bypassing any defensive boosts the target may have. These two categories combined are much smaller than the third category, status moves. Status moves are broadly categorized as any attack that does not directly cause damage, and can buff/debuff yours or your opponents' stats, afflict opponents with passive conditions, or introduce new field effects (such as summoning a downpour or extreme sunlight), to name just a few. Status moves are by far the most important category of attacks in the Battle Maison, since the diversity of Pokemon (and by extension, the myriad critical hits and rare side effects) you may face means that you absolutely want to limit the amount of hits your team members must take.