An attack from a resisted type attack have it's base power halved.
An attack from a weakness type attack have it's base power doubled.
An attack of the same type as the pokémon using it has it's base power multiplied by 1.5

The base stat is determined solely on the pokémons species and is always static, they cannot be influenced in any way, shape or form. This naturally means that some Species are weaker then other species

These are the pokémons "genes" and are determined randomly among wild pokémon. Each value range from 1 to 31 and each point provides 1 extra point in that stat at level 100. It's Possible to affect this value through breeding as your pokémon inherent three random stats total from either of it's parents, certain items makes this passing down garantueed.

This represents your pokémons training and is aquired by fighting, newborn and wild pokémon starts with 0 EV's.
For every 4 EV's your pokémon gains 1 extra point in that specific stat at lvl 100, you may have up to 252 EV's in a single stat up to a total of 510.

EV's are aquired at a static rate depending on the species of the pokémon you defeat, each time you gain XP from defeating an opponent pokémon (wild or trained) you gain 1 to 2 EV points depending on the specie defeated (there's charts that can tell you what specie gives what, general rule of thumb. If a pokémon excels in an area it gives that stat)
Certain items, drugs and virae (yes really) can speed up this process

Each pokémon possesses a nature that influences it's total stats after Base stats, IV's and EV's are added together. Each nature gives a 10% bonus to one stat and a 10% penalty to another, there's 25 natures in total one for each combination and five neutral ones.
Nature is determined randomly but can be influenced to reflect a parents nature if the parent is holding a certain item during breeding.
As you can imagine nature can mean a world of difference.

Health points (HP)
Determines the total hp of your pokémon.

Attack (Att)
Added together with other modifiers to determine the total damage of physical attacks

Defense (Def)
Detracts from the power of physical attacks

Special attack (SpA)
Same as attack but for special attacks

Special defense (SpD)
Same as defense but for special attacks

Speed (Sp)
Determines which pokémon goes first in a turn, the pokémon with the highest speed stat always go first, unless a priority move is used.

Each pokémon possesses a passive trait known as an ability, it's determined by the species and can produce a number of effects, some positive other negative, some are double-edged swords. If a specie possesses several abilities it's equally likely to have either, with the exception of hidden abilities that can only be found in the dream world.
Abilities can include everything from summoning sandstorms on the battlefield, getting stat boosts when a certain condition is met or lowering your stats when a condition is met.
Abilities can make or break a pokémon, even more so then base stats

Every move possess a base power (except non-attacking moves such as stat boosts or status inflicters) a type, accuracy and is either physical or special.

Base power: This determines the damage of an attack before modified by stats, items, abilities or STAB.
Type: this determines element associated with the move, determining how it interacts with weaknesses, resistances and STAB.
Accuracy: determines the chance to hit your opponent even 100% accurate moves have a 1 in a 255 chance to miss (an old glitch that was never rectified)*
Physical or Special: This simply determine whether the move is modified by your Attack stat or Special Attack stat.

*Accuracy can be affected by accuracy or evasion moves. Evasion moves are illegal in competitive play.

Some moves provide secondary effects such as but not limited to

Status inflictions:
Affects the opponent with one of the following effects.
NOTE: A pokémon can only be affected by one status at a time
Poison: deals 1/16th of the opponents total health at the end of each turn, badly poisoning deals exponentially more damage with each turn.
(poison and steel types can't be poisoned)
Burn: Deal 1/16th of the opponents health at the each turn and halves the opponents attack stat. (Fire types can't be burnt)
Paralysis: Cut the opponents opponents speed down to 1/4 and gives a 25% chance to prevent the opponent from acting that turn (ground and electric types can't be paralyzed by most effects)
Freeze: Opponent can't act until thawed out by random chance or using a fire attack
Sleep: sleeping pokémon can't act with the exception of some moves designed to be used only when sleeping, a sleeping pokémon wakes up after 2-5 turns.

Secondary status:
NOTE: A pokémon can be affected by several secondary statuses at a time.
All these effects can be cured by switching out.
Confusion: gives the opponent a 25% chance of attacking himself instead of the opponent
Flinching: prevents the opponent from acting that turn, only effective vs slower opponents
Charm: Gives an opposite gender pokémon a 25% chance to not be able to act that turn
Leech seed: Drains 1/16th of the opponents health and give to yourself at the end of each turn
Taunt: Prevents the opponent from using non-attacking moves
Curse: deals 1/4 of the opponents health at the end of each turn

Hazards: these are semi permanent effects that damages or statuses a pokémon entering the field.

Priority moves:
These moves either act before or after an opponent regardless of speed, the exception being if the opponent also uses a priority move. Some priority moves requires certain conditions to be met, like being the first turn the pokémon is on the field or the opponent must use a attacking move.

Stat changers.
These moves increase or decrease one or more stats of a pokémon by 1 to 3 stages.
1 stage is x1.5 for positive boosts and x0.75 for negative ones.
Each stat can be increases up to 6 stages or reduced to 6 negative stages.

Weather inducers:
These moves sets the weather on the stage for 5 turns, providing benefits or penalties to certain types and moves.

Recovery moves.
These moves heals the pokémon or an ally from damage or statuses.

Critical hits: Critical hits, deal double damage and ignores any modifiers to defense, some moves have a high crit chance

There's plenty of other modifiers to moves such as damaging recoil or charge up turns but these are some of the major ones