Gameplay of Pokémon

The gameplay of the Pokémon series of role-playing video games involves the capture and training of a variety of fictional creatures called "Pokémon" and using them to battle other trainers. Every generation of games follows this concept by introducing new Pokémon, items, and gameplay concepts. Some of the general concepts were featured elsewhere before being introduced in the games; double battles appeared in the anime long before appearing in the games, and Pokémon abilities are similar to the Pokémon Powers introduced in the "Pokémon Trading Card Game".
Battles.
Battling is a large part of the games. Pokémon can be found in the wild, in dark grass, caves, forests, ponds, the ocean, and even underwater. Also, whenever the player walks where another trainer can see the player, the two must battle. Hundreds of trainers are in each of the main series "Pokémon" games.
Statistics.
Pokémon have six statistics (HP, Attack, Defense, Special Attack, Special Defense, and Speed) which all affect how they perform in battles. Statistics can be raised by using items and by training the Pokémon.
Before "Pokémon Gold and Silver" were released, the "Special Attack" and "Special Defense" stats were combined into one statistic, "Special." This later became too complicated so it was divided into "Special Attack" and "Special Defense."
Abilities.
Starting in "Pokémon Ruby and Sapphire", Pokémon have abilities which affect them in various ways. These can be abilities that work while in battle or when the player is just walking around. An example is the ability "Overgrow" which increases the power of Grass-type moves when a Pokémon's health is below 1/3 of its full amount.
Pokémon types.
Each Pokémon has either one or two types. Which types it has affect how it performs when it battles other Pokémon of other types. For example, a Fire-type Pokémon is weak against Water-type, but is strong against a Grass-type. It could be compared to a more complex game of rock, paper, scissors.
This is a list of the types:
Two more were added when "Pokémon Gold and Silver" were released:
Another type, "Fairy", was added in "Pokémon X and Y".
Pokémon Gyms.
Pokémon Gyms are buildings where the player faces a Gym Leader, who focuses one of the 18 types of Pokémon. There are also trainers in the Gym that the player can battle, and they also use mostly that type. Beating the Gym Leader earns you a Badge which increases your skills.
Evolution.
Pokémon evolve to turn into different Pokémon from the same evolutionary line. Doing so, they become stronger and can learn more powerful moves, but learn moves at higher levels.
Evolution by leveling up.
This is the most common way that Pokémon evolve. Pokémon in this category evolve when they reach a certain level. 
Some Pokémon can only evolve by level in a certain place.
Some Pokémon evolve by level if it is holding a certain item.
Some Pokemon only evolve by level if they are leveled in a route that is having a certain weather. This weather can only work to evolve the Pokemon if the weather is not caused by a move or Ability.
Evolution by stones.
This is another way that Pokémon evolve. Some Pokémon can only evolve if they are given certain evolution stones.
Evolution by trading.
This is another way that Pokémon evolve. Some Pokémon only evolve if they are traded to a friend. 
Some Pokémon only evolve if traded to a friend while holding an item.
Some Pokemon only evolve if they're traded with another, specific Pokemon.
Evolution by happiness.
Another way Pokémon evolve is through happiness. Whenever the player character in the game walks 100 steps, gives their Pokémon a vitamin such as Protein, Iron, Carbos, Calcium, or Zinc, training it until it gains a level, or when the character does something else to make it happy, it will gain happiness points. The item Soothe Bell multiplies this amount. Once a Pokémon has reached a certain happiness level, it will evolve after it is leveled once, sometimes depending on the time of day:
Example: Eevee evolves into either Espeon or Umbreon if its happiness rating is maxed out; into Espeon if this happens during the day, or Umbreon if it happens during the night. However, such day-and-night distinction exists only in the second-generation games and the generations that follow (but not in "Pokémon FireRed and LeafGreen").
Evolution by moves.
Pokémon can evolve after learning certain moves and leveling up.
Evolution by gender.
Some Pokémon can evolve only if they are a certain gender.
Mega Evolution.
Some Pokémon can evolve into a Mega form during a battle if they are holding a Mega Stone. However, the evolution only lasts for the battle.
Items.
A large number of different items exist. They are used to help the player or their Pokémon.
Poké Ball.
A Poké Ball is a red-and-white sphere that is used to catch Pokémon. Many different forms of it exist, such as the Great Ball, the Ultra Ball, the Heavy Ball, the Net Ball, the Dive Ball, the Friend Ball, and the Master Ball (which can catch any Pokémon without fail). Each of these have different colors and can be used to catch different Pokémon more effectively.
Pokédex.
The Pokédex is a tool used by Pokémon trainers all around the Pokémon world in different regions. These regions include the Unova region, the Sinnoh region, the Hoenn region, the Johto region and the Kanto region. The Pokédex makes records of information about each of the different Pokémon. There are 801 different species of Pokémon recorded in the world today. Different Pokémon are caught in different regions and some Pokémon can only be caught in certain regions, such as Xerneas in the Kalos Region.
Rare Pokémon.
Legendary Pokémon.
Legendary Pokémon are Pokémon that are very rare, and there is usually only one in each game. Examples of legendaries are Articuno, Zapdos, Mewtwo, Lugia, Groudon, and Dialga. They are hard to catch, and are stronger than most Pokémon. Some roam around, while others can be found in one spot until you KO or capture them. If you KO a legendary, it will usually leave and will never be found again unless the game is restarted or the Elite Four is defeated again. Because of this, players usually save before engaging the Pokémon.
Mythical Pokémon.
Mythical Pokémon are the rarest type of Pokémon, only found by one-time events. Examples of Mythical Pokémon are Mew, Celebi, and Jirachi.
Shiny Pokémon.
Shiny Pokémon are a different color than that Pokémon normal for their species. Some examples would be a yellow Bulbasaur (they are usually greenish-blue) or a red Gyarados (they are usually blue). They are extremely rare; there is only a 1 in 4,096 chance of an encountered Pokémon being shiny, but there are ways to improve your chances of obtaining one, which vary per game. Some Pokemon in the same evolution line have different shiny colors, even if their regular colors are the same. For example, Ponyta and Rapidash normally have reddish-orange flames. However, shiny Ponyta have blue flames while shiny Rapidash have silver flames.