The canonical Kakuro puzzle
is played in a grid of filled and barred cells, "black" and "white" respectively. Apart from the top row and leftmost column which are entirely black, the grid is divided into "entries" -- lines of white cells -- by the black cells. The black cells contain a diagonal slash from upper-left to lower-right and a number in one or both halves, such that each horizontal entry has a number in the black half-cell to its immediate left and each vertical entry has a number in the black half-cell immediately above it. These numbers, borrowing crossword terminology, are commonly called "clues".
The object of the puzzle is to insert a digit from 1 to 9 inclusive into each white cell such that the sum of the numbers in each entry matches the clue associated with it and that no digit is duplicated in any entry. It is that lack of duplication that makes creating Kakuro puzzles with unique solutions possible, and which means solving a Kakuro puzzle involves investigating combinations more, compared to Sudoku in which the focus is on permutations. There is an unwritten rule for making Kakuro puzzles that each clue must have at least two numbers that add up to it. This is because including one number is mathematically trivial when solving Kakuro puzzles; one can simply disregard the number entirely and subtract it from the clue it indicates.
In discussing Kakuro puzzles and tactics, the typical shorthand for referring to an entry is "(clue, in numerals)-in-(number of cells in entry, spelled out)", such as "16-in-two" and "25-in-five".
Rules adapted from content available at Wikipedia. Licensed under the GNU Free Documentation License.