Fudging is reporting the result of a dice roll, made in secret (e.g. behind a GM screen), to be something other than what it was. This is generally only considered to be fudging when done by the GM, and when done by any other player it is considered cheating (although some rare groups may tolerate this).
Even when done by a GM, fudging is a controversial issue.
Why fudge a die roll?
The most common reasons for a GM to fudge a die roll are to improve the story, protect the player characters (e.g. to prevent unexpected character deaths or even TPKs), to adjust encounters in which the statistics are not appropriately balanced, to favor popular non-player characters, or otherwise to direct a particular outcome they think is dramatically appropriate.
Supporters of fudging die rolls argue that doing so is one of the GM's tools for curating the experience of the player group, and that fudging die rolls occasionally can make play more fun.
Can the GM cheat?
In some conceptions of the GM role, it is impossible to cheat. The GM controls the universe, including the significance of any die roll. Hence, changing a die roll to suit some greater purpose is just part of their job. This may be done for dramatic reasons, to increase fun, or to be kind to a player who has had a run of bad luck.
On the other hand, other people consider GM die rolls to be just as important as player rolls in creating genuine tension and chance in a game. In this view, the GM is free to set ground rules, but they must abide by those rules, including results suggested by the dice. GMs should be advised that many players feel strongly about this viewpoint. If they discover the GM has been fudging, they may become disillusioned or upset.
Fudging is a patch for inappropriate game rules
For example, if a group primarily desires a good story in their gaming, rules might be inappropriate if they have a high chance of uninteresting, undramatic or disappointing outcomes. This may be because the results of the game's random resolution have too much swinginess, meaning the range of possible results is large and the probability of extreme results is high. For example, a d20 resolution mechanic will have a range of 19 points, and each extreme result (a natural 1 or natural 20) happens 5% of the time; however, a 4dF resolution mechanic has a range of only 7 points, and each extreme result (-4 or +4) happens only 1.23% of the time, meaning it is less swingy. Fortune at the end mechanics (which do not give players any options to influence undesirable results, e.g. through player resources) may be less desirable than fortune in the middle mechanics (which do).
Dice should not have been rolled in the first place
Another criticism of fudging is that it is only necessary if the actual outcome produced by a roll is undesirable to the GM and the group as a whole, and dice rolls should only be made in the first place unless success and failure are both interesting. According to this principle, judgemental resolution should be used in other situations to ensure that the interesting outcome takes place, and that this should not be hidden from the players.
Fudging to disguise fiat
Sometimes, a GM will roll a die, even though they have already decided on something. The GM may reserve their use of the GM fiat for dramatic or game purposes, allowing the dice to dictate how a scenario plays out. In this case, the GM is not really fudging a die, but engaging in GM obfuscation. If this is done frequently, players may (justifiably) complain of railroading.