A constraint is a rule that limits which digits can be placed in a cell. Standard Sudoku has three constraints - row, column, and box - and every solving technique is a method for combining them.
A constraint is the formal name for one of Sudoku's "no repeats" rules. Standard Sudoku has three: each row, each column, and each box must contain the digits 1 through 9 exactly once. That is a total of 27 constraint groups (9 + 9 + 9), and every cell belongs to exactly three of them.
Every solving technique - from naked singles to forcing chains - is just clever bookkeeping over these constraints. Eliminations remove digits forbidden by a constraint; placements satisfy three constraints at once.
When you place a digit in a cell, that placement broadcasts to the rest of the cell's row, column, and box. Every other cell in those three constraint groups loses that digit as a candidate. This cascade is what turns one placement into many.
Constraints also let you reason in reverse: if a digit appears in only one cell within a constraint group, that cell must hold the digit (a hidden single). If n cells inside a group share n candidates, those digits are locked there (a subset).
Sudoku variants are defined by their extra constraints. Diagonal Sudoku adds two more 'no repeat' lines. Killer Sudoku adds cage-sum constraints. Thermo Sudoku adds increasing-order constraints along thermometers. The solving logic is the same - more constraints, more eliminations.
Effectively yes. 'Constraint' is the precise term used in puzzle theory, while 'rule' is the everyday word. Both refer to the requirement that digits 1-9 appear exactly once in each row, column, and box.
Three in standard Sudoku - one row, one column, one box. In variants like Diagonal Sudoku, the four cells on each diagonal also touch a fourth constraint.
That would just be a Latin square - far easier than Sudoku and usually with many solutions. The box constraint is what makes Sudoku interesting and gives it a unique solution.
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