Two moves build one. Push the knight pawn a single square, g2-g3 or b2-b3, then slide the bishop into the hole it left, Bf1-g2 or Bc1-b2. That bishop now looks along the longest line on the board.
Here it is from Black’s side, in a King’s Indian Defence: 1.d4 Nf6 2.c4 g6 3.Nc3 Bg7, and the bishop on g7 is pointed at d4, c3, b2 and a1.
8 
7 
2 
a 
c 
d 
e f g h 1 The diagonal is the point
A bishop on f1 has to wait for the e-pawn to move and then usually lands on c4 or e2, where it hits one or two useful squares. A bishop on g2 hits eight, from g2 all the way to a8, and it keeps hitting them for the whole game because the pawn on g3 cannot be blocked away in front of it. In the Catalan the g2 bishop leans on d5 and c6 from move three and is still doing it forty moves later.
That permanence is why the fianchettoed bishop is so often the best piece on the board even when it never moves again.
What it costs
One tempo, first of all: two moves for a development that plain Bc4 does in one. And a commitment. The pawn on g3 is a small hole under g2, and the dark squares f6, g7 and h6 around a castled king are guarded by exactly one piece. Trade that bishop off, or lose it, and those squares belong to whoever brings a queen near them. This is why strong players are happy to spend a bishop and several moves getting rid of a fianchettoed one, with Bh6 in the King’s Indian for instance.
The rule I give students: if your fianchettoed bishop leaves the board, be careful about the colour it was standing on.
Where you will meet it
Black fianchettos on g7 in the King’s Indian, the Grünfeld, the Sicilian Dragon and the Pirc. White fianchettos on g2 in the Catalan and the English, and on b2 in the Nimzo-Larsen with 1.b3. On both wings at once is a hypermodern setup that hands the centre over on purpose and shoots at it from the corners.
Play a few kingside attack puzzles against a fianchettoed king and you will see the pattern from the other side: the attacker’s first idea is nearly always to swap that bishop off.