Is a Gish a class within Dungeons & Dragons or any race that focuses on both magic and combat skills at the same time? How would one go about creating a Gish and what are the pros and cons of doing so?
I've attempted to make a few of these over the years, the character building is quite involved but fun. The Gish is a term originating from the Githyanki - a race that focused on combining fighting with magic - but not all Gish are Gith. In Advanced Dungeons & Dragons building a multi-class Fighter/Magic-User isn't the only option, as clerics can cast (Divine) spells and fight too. However, divine and arcane casters have a different repertoire of spells so often a Gish is assumed to use arcane magic. Elves can dual-class as a fighter/mage and that's one option. Githyanki also could, however they came out later.
In 3.5 edition it's possible to build a Gish, but you have to accept you will be versatile rather than a master of sword and spell. You will be outclassed by pure casters but have the ability to have a longer adventuring day, with your weapon as backup. Building one is also a balancing act and you have to plan and pick the right race and classes. Prestige Classes are a huge boon in this edition as you get abilities progressing both your melee and spellcasting.
Pathfinder is similar and also includes the Magus if you want to avoid multiclassing, which keeps its power just under par with the other classes. Many class archetypes also alter classes to include martial abilities at the cost of losing some magical prowess.
4th edition gave powers to all classes, barbarians communicated with nature spirits, monks had psionic powers, and classes like the warmage and warpriest were built to combine spells with combat. Unless you were a pure spellcaster or fighter, in my opinion you are a Gish.
Conversely, 5th edition opens up archetypes for all classes. You can specialise into something resembling a Gish, like the Elemental Knight, Way of the Shadow Monk, or Arcane Trickster, etc. However each of those are quite different from each other so it's hard to group them together as Gish.
As Icon hints at, the problems faced can vary depending on the edition.
But in most cases, it boils down to a question of what you give up and to what degree. An archmage that can avoid stabbing themselves with a sword isn't tough to manage, and a blademaster that knows a handful of magic tricks isn't tough either. Where it gets difficult is in being good enough at both things to not be laughably behind in at least one of them.
If however, one succeeds to a reasonable degree, a gish can be a very capable character, combining spells and physical might in a potent combination.