Poké Ball

A Poké Ball (Monster Ball in Japanese) is a small mechanical ball used to catch Pokémon. There are numerous varieties of Poké Balls, but the most basic one is simply referred to as a Poké Ball. The Poké Ball is also the symbol of the Pokémon series, similar to Zelda's Triforce or Mario's Super Mushroom.

The function of Poké Balls is not only to catch Pokémon, but as a very easy and convenient means of carrying them around with you (without having to use items such as leash, which is convenient as some Pokémon can travel on land easily or by their size). Poké Balls are small, perhaps the size of a golf ball, for easy storage and carrying. When a white button is pressed along the seam of the ball's halves, however, it will expand to about the side of a softball, making it easier to throw and release the Pokémon inside.

When a Poké Ball is thrown at a Pokémon during a capture attempt, or when releasing/withdrawing a trainer's Pokémon, the Pokémon becomes a blob of red light/energy. This is never fully explained, but Pokémon shrink or compress in some way in order to fit into Poké Balls. Poké Balls can also contain items, which are found laying around in forests and caves, though the ball is presumably used up when the item is removed.

While trainers can carry as many empty Poké Balls as they want, they can only carry a max of 6 Poké Balls containing Pokémon at one time, according to the rules of the Pokémon League. When a trainer has caught more than 6 Pokémon, the Pokémon after that will be automatically transferred to PC Storage.

Catching Pokémon
To catch a Pokémon, you first need a Pokémon. (It is presumed that most people buy their first Pokémon or are given one.) The Pokémon you are trying to catch must also be wild, and not already owned by another trainer, as the Trainer will block the ball. The effectiveness of a Poké Ball is determined by how it changes a Pokémon's chance of being caught.

Catch rate is determined foremost by a Pokémon's total HP, remaining HP, and the innate catch rate of the Pokémon you are trying to catch. Weak Pokémon such as Caterpie have a catch rate of 255. Powerful legendary Pokémon such as Mewtwo have a catch rate of 3, and are nearly impossible to catch without the use of a Master Ball.

When you throw a Poké Ball at a wild Pokémon, the game uses the following formula to determine whether it works or not. 

$$Chance = floor[{(Total HP*4 - Current HP*2) * Innate Catch Rate \over Total HP}] + Status + 1$$


 * Note: "floor" means that the number is always rounded down, even if it ends in something like ".99999".
 * If Current HP < 255, use floor(Current HP/4). Total HP works the same way, but with a Pokémon's max HP.
 * If a Pokémon is asleep or frozen, status = 10. If it is paralyzed, poisoned or burnt, status = 5. Otherwise it equals zero.

After all of this has been calculated, the game generates a random number between 0 and 255. If the random number found in the calculations is smaller than the number calculated in the above calculation, the Pokémon is caught. So the higher the number, the better your chance of catching the Pokémon.

The role of Poké Balls is to add a multiplier to the above equation. See the graph below for each Poké Ball's multiplier.

In short
Basically, players weaken the Pokémon to a certain extent and use the most effective ball available to catch it. If in some case the ball fails, players can try again and hope for a better chance to catch the Pokémon.

General
The following Poké Balls are in all Pokémon games:

Gold/Silver/Crystal Poké Balls
You get these balls by giving Apricots to Kurt.


 * Fast Ball - If the Pokémon tends to flee from battle, this ball has a 4x multiplier. Has an effect on Magnemite, Grimer, Tangela, Mr. Mime, Eevee, Dratini and Dragonair.
 * Level Ball - If your level is higher than the wild Pokémon's, 2x. If twice as high, 4x. Four times or higher, 8x.
 * Heavy Ball - If the wild Pokémon's weight is between 100 kg and 200 kg, 2x. If it's between 200 kg and 300 kg, then it's plus 20. More than 300 kg, +30. Under 100 kg, -20.
 * Lure Ball - If wild Pokémon is fished out of the water, 3x.
 * Friend Ball - Same as a normal Poké Ball. However, the Pokémon will start with a tameness/happiness value of 200.
 * Moon Ball - If wild Pokémon can evolve from a Moon Stone, 4x.
 * Love Ball - If the wild Pokémon is a different gender than the one you're fighting it with, 8x.

Ruby/Sapphire/Emerald Poké Balls
Pokéball's introduced into the Ruby/Sapphire games (and subsequently used in Emerald, Leaf Green and Fire Red and Colosseum and Pokémon XD versions)

Diamond/Pearl Poké Balls
Two new Poké Balls for Pokémon Diamond and Pearl have been unveiled so far. It is also assumed that R/S/E Poké Balls will be available as well, at the very least by attaching them to your Pokémon before transferring them over via the Nintendo DS GBA slot.