In a nonogram the numbers are not something you fill in — they sit along the edges and describe the pattern you are uncovering. A row marked 4 2 contains four filled cells in a row, then at least one gap, then two more. The order is fixed; where each group starts is what you have to work out.
How to solve it
The essential technique is called overlap, and it feels contradictory until you see it. Picture a row of ten cells with a single clue of 8. You do not know where the run begins, but it can only start at cell one, two or three. Push the run as far left as it goes, then as far right, and compare: six cells in the middle are filled either way. You can fill those immediately, knowing nothing else.
Mark empty cells as diligently as you fill full ones. Cells you know are blank are not wasted information — they are what constrains where the runs can sit, and most people who get stuck have stopped marking them.
Work the longest clues first, and the rows and columns that are nearly complete. Every newly filled cell tightens the constraints in the other direction, so keep alternating between rows and columns.
What makes a puzzle hard
Size does most of the work. A five by five grid fits in your head; fifteen by fifteen forces you to track clues you cannot see at the same time. Many short runs are also harder than a few long ones, because the overlap technique pays less the shorter the run.
A worked example
Take a row of ten cells with the clues 3 4. The runs need three plus four cells and at least one gap between them, so eight of the ten are spoken for. That leaves only three possible placements, and comparing them shows that cell three, and cells six and seven, are filled in every one. Three certainties, with nothing else known about the row.
The most common mistake
Trying to guess where the pattern is heading. The shape emerges as a by-product of the logic, and chasing a form you think you recognise almost always goes wrong. Trust the clues, not the silhouette.