Godmodding is a wide type of issue that encompasses many different things. One of the major aspects is that Godmodding enhances their characters abilities beyond what they're capable of normally. This means they are suddenly stronger or faster than they realistically ever were (without valid reason, see exceptions) or have new unheard of powers (especially when having new power suits them… Like if they suddenly needed a fire attack they'll whip one out despite not actually having one.)
Though godmodding only usually affects the player's character directly (unlike powerplaying). Though it can overlap with other issues (like for players that use out of character knowledge to find weaknesses so they can create new special moves just for that fight).
Godmodding has several major issues that cause detriment to games, these being:
Godmodding takes any challenge or struggle out of that situation. It makes everything too easy. Having easy gets extremely boring, very quickly. (Which is why characters who are over-powered are generally boring, because they never struggle, they never fail, they never falter and their audience can't easily relate to them. Compare Superman to Batman… Batman is human, fails and struggles which makes him a more relatable character than Superman who can power through most situations.)
People often don't like when others godmod. Like previously mentioned it gets boring and predictable. But when godmodding is used against players it creates frustration. Players can't deal with godmodding. It removes any sense of fair play and in many cases godmodding causes other players to counter-godmod (or godmod in retaliation). Which creates a vicious cycle of godmodding.
Staff may end up having to be called in to resolve situations since godmodding is usually against the rules of the forum/roleplay. And no admin really wants to have to tell a member that they can't do something, it makes the admin feel like the villain. (Even if they're in the right in telling the member to behave.) Also, no member likes to be told that their behaviour is unacceptable. They may resent the admin for telling them to behave or at least feel like they're being unduly picked on.
Godmodding is relatively easy to avoid if players:
If they're not sure about an action they want to take, they can always ask another player if they'd have an issue if it were done against them. (Since other players wouldn't like having someone godmod against them.)
Not every change in power is godmodding though. Characters, over time, may naturally grow stronger (especially in roleplays that revolve around growing more powerful). So a character at the start of the roleplay (or when they join) may well be weaker than later on. Similarly, adding new moves over time may not be godmodding. Characters often learn and evolve over time to change their arsenal/powerset up. (Though this will depend on the roleplay in question since in some roleplays characters may not change during it's duration if the entire roleplay happens within the span of a week.)
Though sudden increases in power (or new abilities) alone doesn't necessarily indicate godmodding, the story may give new abilities or increase powers temporarily as part of the story. (Consider in inFamous2 where Cole McGrath draws lightning from a storm to power up. ) The difference though is the rate of growth or change. Godmodding is instant growth with no explanation (just a case of sudden power increase or a sudden new move) for just one person because they want more power right then.