The Final Fantasy series isn't known for having a consistent continuity -- every mainline entry has been a standalone story with its own fantastical world and cast of colorful characters. While there have been sequels and spin-offs to individual entries, such as Final Fantasy X-2 or Lightning Returns, there has never been any kind of concrete connection between any of the games beyond sharing some magic spells, summons and monsters.
In Final Fantasy, even characters who share the same name (such as Cid) are wildly different from one another. However, there is one character who has remained relatively consistent in appearance and personality, having appeared in several games in the series as both an enemy and an ally: Gilgamesh.
Typically, Gilgamesh is a wandering sword collector who is constantly searching for the holy sword Excalibur, but often, he ends up with its vastly inferior duplicate, Excalipoor. He is a very boisterous and comical man, loudly bragging about his strength in a dramatic manner. However, he also often clumsily falls over himself, tells terrible jokes and is easily tricked considering how often he mistakes Excalipoor for Excalibur. Still, players shouldn't let his apparent ineptitude fool them -- Gilgamesh is at least almost as strong as he claims to be, wielding an arsenal of collected swords at his disposal and capable of powerful Jump attacks.
Since debuting as a boss in Final Fantasy V, Gilgamesh has appeared in many Final Fantasy titles, sometimes as a boss, sometimes as an ally and sometimes as a summon. But initially, it was not entirely certain whether every version of Gilgamesh that appears is in fact the same person, as some games (including his debut) appear to kill him off permanently for one game.
The Dissidia games, however, heavily imply that the Gilgamesh of other games and the original Gilgamesh from Final Fantasy V are one and the same. This Gilgamesh possesses both abilities he has shown in later games, as well as a burning desire to fight FFV protagonist Bartz Klauser, his original rival. With the Dissidia games revealing that it's possible for characters to cross universes, Gilgamesh may be the one character tying all the games together.
While seemingly non-canon, the chibi-fied spin-off World of Final Fantasy reveals in a character profile that Gilgamesh does in fact have the ability to cross dimensions. He's said to have "a very special soul" that is imprinted with his original self's memories of Bartz. This means that even if his body dies in one game, his obsession with Excalibur and Bartz will allow his soul to rematerialize between dimensions, though how he does this is still unknown.
Gilgamesh appears to grow in strength with every game he appears in, whether he appears as a boss, a summon or both. While his appearance changes rather frequently, he continues to maintain a grey complexion and wear thick red/orange cloaks and armor, often possessing multiple arms to wield his many, many blades. Regardless of his role in a game (or whether or not Excalibur is involved) Gilgamesh continues to maintain the hammy attitude that made him a fan-favorite character.