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Bant & Jund are names of the Shards of the once-broken plane of Alara, in the Shards of Alara Block
the others are Esper, Grixis, and Naya, making it 5 Shards, each with its own majority 3 colors
when somebody says "Bant" or "Jund" they're usually referring to the colors in the deck
for example a Blue-Green-White deck would be Bant, as that was Bant's majority colors.
They're references to the third-most recent story arc & world setting, Alara.
Alara was split into five shards along the colors of MtG, each shard only getting three colors of mana and thus only having access to those three "types" of magic. Each shard had a unique and distinct personality as a result.
"Bant," "Esper," "Grixis," "Jund," and "Naya," were each names of one of the shards.
Bant - Green, white, blue
Esper - White, blue, black
Grixis - Blue, black, red
Jund - Black, red, green
Naya - Red, green, white
More commonly, you'll hear them used to refer to a color grouping in a deck. For instance, a deck which uses green, white, and blue will be called a "Bant deck" or "Bant colors" even if it's not really using any Alara cards.
"Jund" in particular has some infamy, as when the Alara block was in Standard there was a particularly terrifying, extremely card-efficient type of deck that used Jund's colors and its special Devour mechanic.
Bant, Jund, Esper, Grixis, and Naya are the five Shards of Alara. Each of them have three colors of Mana.
Bant - U,W,G; Home of Angels
Naya - R,G,W; Home of Gargantuans
Jund - B,R,G; Home of Dragons
Grixis - U,B,R; Home of the Undead
Esper - W,U,B; Home of Artifacts
Naya is my personal Favorite cause of the Huge creatures.