Sudoku is 81 cells arranged in nine boxes of three by three. There is a single rule: every row, every column and every box must contain the digits 1 to 9, once each. No arithmetic is involved. The digits are just symbols, and the puzzle would work equally well with nine colours or nine letters.
How to solve it
Start by scanning. Pick a digit — say 7 — and work through the boxes one at a time. If a neighbouring box already holds a seven, draw imaginary lines through its row and column: no new seven can sit there. Often a single cell survives in the box you are looking at, and that placement is certain.
When scanning dries up, flip the question. Instead of asking where a digit can go, ask what can go in a particular cell. Rule out eight of the nine digits and the ninth is forced.
Notes mode exists for that second question. Pencil every candidate into the empty cells and the patterns surface on their own: two cells in the same row that can only hold 4 and 8 mean no other cell in that row can contain either digit.
What makes a puzzle hard
The number of given digits matters less than people assume. What decides difficulty is which techniques are needed to make progress. An easy puzzle yields to scanning alone. A medium one requires switching between the two questions above. A hard one requires working with candidates and spotting relationships across several cells at once.
The most common mistake
Guessing. A properly constructed Sudoku has exactly one solution and never requires a gamble — every digit can be deduced. If you are stuck, you have missed something rather than met a puzzle that demands luck. Back up and scan again.