These terms will not be in order of importance. I'm going include all the terms I feel are necessary to be able to understand and play the games.
HP- Hit points This shows how much health your Pokemon has. Lose them all and your Pokemon will faint. You can revive them by using the Revive item or bringing them to the nearest Pokemon Center.
PokeBall- This is what you use catch and store the Wild Pokemon you ecnounter and decide to capture. There are 15 kinds added in various games. HeartGold and SoulSilver even allow you make your own special kinds with Apricorns.
Apricorns- Found only in HeartGold, Soulsilver, Gold, Silver and Crystal. Used to make special Pokeballs depending on the color Apricorns used, of which there are 7.
Pokedex- An electronic device that catalogs and shows information regarding Pokemon that you encounter or capture. Whenever a Pokemon is captured, it's height, weight, species and a short description will be added.
Every game and region has their own Pokedex. There is an optional feature to sort the Pokemon. You can even compare the height of the Pokemon to the height player character.
Evolution- Nearly every Pokemon in the game series does this at some point; usually at certain levels or after certain conditions have been met. You can even pre-evolved forms(or baby pokemon basically) of some Pokemon.
Evolving is like a metamorphosis. As the Pokemon evolves there is an increase in stat values. Some Pokemon can only Evolve if they are traded. There is even a fan term called Eeveelution and is related to the evolution of a Pokemon called Eevee.
Stats- There are 6 of them. They are: HP, Attack, Defense, Special Attack, Special Defense and Speed. These stats increase whenever the Pokemon levels up. There are also hidden stats called Accuracy and Evasiveness that you can't do anything by your actions in the game to increase.
There are also moves and items that can either temporarily or permanently increase or decrease the first 6 stats.
Natures- These are given randomly and can't be changed. There are 25 of them. 20 increase one stat and decrease another at the same time. While 5 modify the same stat with an increase and decrease(this is what is called video game logic, meaning it doesn't make sense in the real world).
The natures affect the degree to which the stats increase or decrease. In addition a Pokemons nature determines which treats it will like. A treat can function as a reward, be nutrition. Or just a treat (for fun) that you use to help strengthen the bond with your Pokemon. The 3 types of treats are: PokePuffs, Pokeblock and Poffins.
Items- There are 5 kinds of items: Technical Machines, Hidden Machines, Recovery items, Held items and Key items. TMs and HMs teach moves to Pokemon they would otherwise not be able to learn in the course of normal gameplay.
Technical Machines- Also called TMs; are commonly found in fields, shops and rewards for defeating Gym Leaders. Before the fifth generation of games, these machines could only be used once.
Hidden Machines- Also called HMs; are rare but can be used more than once. These cannot be removed unless you go to a Move Deleter NPC.
Recovery Items- Help the Pokemon heal in and out of battle. There are many kinds of items like potions, antidotes, revives, ethers, elixers and awakenings. Potions recover HP and Revive wakes up and partially heals a fainted Pokemon.
Antidotes helps with certain status effects and Awakening wakes up a sleeping Pokemon. Elixers and Ethers restore PP(Power Points) in varying amounts.
Held Items- Pokemon can only hold a single item. They can do a variety of things depending on the item they are holding, for example berries can either restore a Pokemons' health during battle or boost their stats. Other held items can raise defense or offense, cause Pokemon to evolve or not evolve or can have a permanent effect on stats.
Key Items- These are rarer than regular items. They are given to player instead of being found. These can't be given to any Pokemon, thrown away or sold. Some common items are a bicycle, a dowsing machine, a map, varying quality of fishing rods and keys to important areas.
Generation- Each Game is part of a generation and there are currently 6 of them. There are generally 2-3 games per generation and they tend to bring lots of new Pokemon, new badges, new Gym Leaders, new items and a new region or two.
Gym Leaders- They are basically a boss character. There are 8 in every game. They specialize in battling with a certain Pokemon type.
Badges- There are 8 in every game and they show how powerful you are as a trainer. Getting a certain badge is necessary to make Pokemon up to a certain level obey you. They also give you a unique TM to use and allow you to use the overworld effect of an HM.
Region- The setting of the gam. These are generally based off of actual areas of Japan. Things or places within those areas can sometimes inspire certain Pokemon designs.
PP- Power Points. The amount of times you can use a certain move. Some move can only be used 5 times and others can have alot more. When you run out you can use that move again unless you have an Elixer, Ether or bring your Pokemon to be healed at an Pokemon Center.
Pokemon Center- A place to heal you Pokemon for free and it has a computer that holds all the Pokemon you catch if you don't have space in your party for them. They can also have shops in them that will sell you items.
Berries- They have been in the game since Gold, Silver and Crystal. There a large variety of them(64 in fact) and some are only available in certain games. They can be difficult to get but a feature in Generation 3 introduces 'soft soil' which allows you to plant
some berries and get more of them and at the same time groups berries into families(this refers to 5 berries of the same growth period and similar flavor). They have to be watered frequently or else they won't grow. Just tap on the soil and you find if your berries need to be watered.
Each Genertion introduces more berries and even new berry related mechanics: Generation 6 introduces mutations and compost, Generation 4 introduced a few berries that are event exclusive,
Generation 5 allowed you to grow berries in a Pokemon Dream World(I never played Black, White, Black 2 or White 2, so I don't know much about this mechanic). Some games even have exclusive berries that you can only find in that game.
Starter Pokemon- They are the first Pokemon you see in the game and will always be water type, grass type or fire type. The starters have changed with each Generation and many of them have envolved to be dual-type. Your rival will always choose the Pokemon whose type is super effective against your type.
For example, if you chosoe the grass type starter than your rival will always choose the fire type starter because grass is weak to fire.
Pokemon Gyms- Found throughout the regions of the different Pokemon games. There are 8 gyms in every game. Beat all 8 Gym Leaders(and get all 8 badges) and you will then be able go on to fight and beat(or at least attmept to beat) the Elite Four and The Pokemon League Champion.
Once you have beaten the Elite Four and Pokemon League Champion you have beaten the game. Pokemon Gyms aren't in every town but when they are in a town they are there because you have to beat them to advance the plot and go onto the next area.
Team- A criminal organization in the games that tries to take over the world through misuing Pokemon.
PokeMarts- Where you can buy different items with the money you win during battles.
Other shops- Some cities have other shops like malls or pharmacies. These places sell items you can generally only get in those locations.
Mulch- Introduced in Generation 4. Has no use in battles. It is used with berries to help promote health and growth. In X and Y, you can make your own Mulch with various combinations of berries. In other games the mulch is bought ready made.
The different kinds will have different effects. These effects include: growth mulch that speeds up growing time, boost mulch that increases how many berries you get, gooey mulch which makes a new sprout grow from a dead berry plant and damp mulch which makes the berries glow slowly but keep the soil damper longer.
Legendary Pokemon- They aren't just Pokemon who have legends written about them but are Pokemon that differ from the others in a few important ways. Currently only one member of the group of Legendaries can only be caught. Since Gold and Silver were released there is a title Pokemon that is involved in the story and it is a Legendary.
Legendaries are extremely difficult to catch, exceptionally powerful, cannot be bred and do not evolve. Most are based off mythological and religious figures. There are also Mythical Pokemon who are a subset of Legendary Pokemon.
Shiny Pokemon- Pokemon that look different and are called 'Shiny' by players. This was done to show the color pallete of the Gameboy Color. They are incredibly rare but there are a few exceptions to this. They aren't necessarily better than regular Pokemon but they do have slightly better than average stats.
When the third generation was introduced, the shiny Pokemon weren't better than regular Pokemon and shininess was determined by another value.
Mythical Pokemon- A subset of Legendary Pokemon. They can only be caught through the use of an external mechanic. These are found at the end of their respective games Pokedex and can only be gotten though events distributed by Nintendo.
Type- This is an elemental attribute given to each and every Pokemon. Normal is used when the Pokemon doesn't have an elemental attribute. There are 18 types with Fairy being the newest. All the moves in the games correspond to an elemental type and do more, less or no damage depending on the elemental type(s) of the opponent Pokemon.
Moves- These are what the Pokemon use in battle. Some are gained through leveling up, evolving, using TMs, HMs, breeding or using the Move Tutor. The moves a Pokemon can learn are limited by it elemental type or types.
