Banish, formerly known as being removed from play (often abbreviated as RFG or RFP) and called excluded from the game (ゲームから除外 gēmu kara jogai) in the OCG, is a term used to describe when a card is removed from the Duel by another card's effect. Once a card is banished, it cannot re-enter the current Duel, except by a card that specifically allows it, such as "Return from the Different Dimension" or "Miracle Dig".
Originally, there were not many cards which could banish other cards, with some of the first being "Soul Release" and "Banisher of the Light" (the first card to return them was the card mentioned above, "Miracle Dig"). However, over time, banishing has become a popular theme with several cards to go with it (Chaos and D.D. (Different Dimension) cards). And, in turn, more cards were created to bring them back, including "D.D.M. - Different Dimension Master", "Dimension Fusion", "Dimension Explosion", "Burial from a Different Dimension", and "Leviair the Sea Dragon".
Banishing is good for breaking strategies especially when combined with a card like "Fiber Jar", which reunites every player's hand, Deck, Field Contents, and Graveyard into a new Deck but doesn't combine cards that are banished. Banishing combined with monsters like "Gren Maju Da Eiza", "Golden Homunculus", or any in the "Helios" series can give a player some of the highest ATK in the game.
Since many Decks rely on the Graveyard, Decks that rely on the effects of "Dimensional Fissure", "Macro Cosmos", "Banisher of the Light", and "Banisher of the Radiance", along with "D.D. Survivor" have grown in power. Being able to shut down an opponents Graveyard plus having an 1800 ATK monster that keeps returning itself every time it's banished while face-up for as long as the banishing card is active, have shown themselves to be powerful Decks. "D.D. Scout Plane" is also incredibly useful in these Decks - if it is banished from your hand or Deck, it will be Special Summoned. The main weakness of these Decks is the fact that it's not very hard to get rid of the card that banishes other cards, one card aimed specifically at them is "Imperial Iron Wall".
The Banished Zone can be placed anywhere on the table, as it has no actual place in the game (as demonstrated from its absence on official play mats). However, most players typically placed the "Banished Zone" either above or to the right of the Graveyard Zone, placed face down beneath the Graveyard, or just under the field (right in front of the player). Most video games implement it to the right of the Graveyard.
Thematically, cards which banish tend to either imply that the card's soul is being removed ("Kycoo the Ghost Destroyer", "Bazoo the Soul Eater", etc.) or that the Banished Zone is another dimension entirely (as many cards of the "Different Dimension" series) imply.