The Bogo-Indian arises after 1.d4 Nf6 2.c4 e6 3.Nf3 Bb4+ (ECO E11). The check from b4 is a Nimzo-Indian idea: Black creates pressure at once and is ready either to trade the bishop or to step back with it, having developed quickly.
The opening is regarded as reliable and not overly theoretical: Black gets a solid position with no weaknesses. White usually blocks the check with Bd2 or Nbd2, after which the play is quiet and positional.
The defence is named after Efim Bogoljubov. A convenient choice for anyone who wants the Nimzo-Indian ideas without its sharpest and most heavily analysed lines.
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It is the mating side to move. Find mate in one.
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