Match-3
Also said as Match three, Tile matching, Swap puzzle
Swap two neighbouring pieces to line up three of a kind. They vanish, the pieces above fall, and new lines may form by themselves. It promises small and constant rewards, plus sudden lucky cascades.
Cascades are the secret: they are free rewards you did not plan for. The number of colours is a real design dial, since fewer colours mean more accidental matches.
Say it
“Use six colours on an 8 by 8 board. Matching four makes a line-clearing piece, matching five makes a colour bomb, and raise the pitch a step on every cascade.”
Where you’ve seen it
Eugene Alemzhin's Shariki (1994) came first, PopCap's Bejeweled (2001) made it huge, and King's Candy Crush Saga (2012) added a limit on lives and a map of hand-made levels, turning it into a free-to-play business.
Not to be confused with tetris-like
In Tetris you place falling pieces against gravity and the clock. In a match-3 the pieces are already there, and you rearrange them.