Split the grid into rectangles so each holds exactly one number, and that number is its area.
The full guide, with a worked example and common mistakes: How to play Rectangles.
Yes — Shikaku is the Japanese name for the same puzzle.
Multiplication facts and factor pairs (which widths and heights make each number), area, and checking with addition: all the numbers together add up to the size of the grid.
Yes. A 4 can be a 2×2 square, a 9 can be 3×3.
Open the Rectangles Solver, choose the rows and columns, click each numbered square and type its number, then press Solve.