Min-maxing is the character-building strategy of maximizing a specific desirable ability, skill, or other power of a character and minimizing everything else, seen as undesirable. The result is a character who is excessively powerful in one particular way, but exceedingly weak in others.
Min-maxing has a history of controversy among players and game designers. Game designers may dislike min-maxing because it discourages variety in play through extreme specialization. It can also 'break' the difficulty balance of a game--making parts of a game too easy or too hard--since games are usually tuned with the goal of providing a reasonable (and thus enjoyable) level of challenge throughout for all normal character builds. A min-maxed character build can often puncture the intended equilibrium of difficulty by being unreasonably good at one thing and unreasonable bad at many others.
Furthermore, if the one thing that a min-maxed character is good at is overall more useful (e.g. combat) than other character abilities (e.g. talking or environmental exploration), the player is likely to rely heavily on that one thing they're good at to solve all situations in the game (e.g. killing everyone instead of talking to them). Game designers often attempt to limit the success of min-maxing by including challenges in their games that cannot all be met by any one specialized character build or by incorporating limits into the rules of character building to prevent overspecialization (e.g. point costs to raise an attribute increase the higher the attribute is, or a character's highest level skill cannot be increased more than 5 levels above their lowest, etc.).
Game designers may also dislike min-maxing by players if it means the player sees their character in starkly mechanical terms rather than as a fictional person. As a result, a min-maxing player may be less likely to roleplay their character or to engage with the game's story or other characters in a way reasonable for an imagined inhabitant of the game world.
Historically, min-maxing arose in Dungeons & Dragons and other tabletop games. In roleplaying games like D&D, min-maxing carries a pejorative edge to some players and to many Dungeon Masters. This is because of the reasons mentioned above, with a min-maxed character potentially vastly overshadowing their peers in their area of expertise, being poorly suited to roleplaying, and disrupting the game's balance. On the other hand, to players who primarily see and enjoy games as creative optimization problems, min-maxing is the only fun way to play games.
Over time, videogames have grown more accepting of min-maxing in general and game design has improved to better account for min-maxing by players. Games such as Diablo III, embrace min-maxing as a core part of its gameplay loop. Other games with more story and roleplaying, like Mass Effect, have separate character building systems for in-combat (skill points) and out-of-combat (paragon/renegade points), so that a min-maxed character does not have to sacrifice one for the other.