Many golf tournaments employ a cut that trims the field down to only the top scorers at a certain point in the tournament. After two rounds of a 4-round tournament, for example, the field might be cut, with the bottom half going home and the top half continuing through completion of the tournament.
The cut line is the score that players must have in order to continue playing. For example, if the cut line is +4, then all golfers in the tournament who are at +4 or better continue; those worse than +4 are cut from the field.
That specific number is not known prior to the start of the tournament - only the cut rule used in the tournament is known. On the European Tour, the cut rule is that the Top 65 players plus ties advance; those players outside the Top 65 are cut. So in this example, the cut line is the score that gets the player inside the Top 65 plus ties.
So the cut line is a fluid number that changes depending on how well, or poorly, the field as a whole is scoring. At the midway point of the second round, it might appear that +3 will be the cut line; but if players on the course start making a lot of birdies or a lot of bogies, that number might move in either direction, higher or lower. The cut line can change to +2 or +4 or some other number. The cut rule stays the same; but the specific score it takes to make the cut - the cut line - changes depending on scores being posted by the players.
For some specific cut rules, see:Golf Glossary index
"The cutline just moved from 4-over to 3-over."