Is there a Relationship between the Final Fantasy games?

Strictly speaking, NO, there is no explicit continuity within the Final Fantasy series, except where explicitly noted. (I.E. 12 inhabits the same world as Tactics IIRC, and of course, direct sequels like X-2.)

However, the series does employ many similar themes throughout, in addition to a relatively static (though ever growing!) stable of monsters, summoned monsters and spell types. In-game cosmology, while also variable is often similar. Additionally, the writers make an effort to ensure that certain 'recurring characters' (in name only), such as Cid, Biggs, and others show up in most, if not all games of the series.

Whilst there isn't an explicit relationship between every single Final Fantasy game in the series. Many are linked to each other:

• Final Fantasy II & Final Fantasy IV: In FFII there is a young boy called Kain who is the son of Ricard Highwind. He wants to one day become a dragoon. In FFIV, one of the main characters is Kain Highwind. Obviously, this could just be a coincidence BUT in the DS version of FFIV he also mentions that his father was called Ricard and he died fighting an evil empire. Moreover, in a Japanese guide for FF4 its mentioned that Cecil's Deathbringer once belonged to a dark knight called Leonhart who was indeed, a character in FFII. They also both have a town called Mysidia which is full of lovely little mages.

• Final Fantasy IV & Final Fantasy VI:

  General evidence to support this largely includes the Ancient Castle. In Final Fantasy IV, Baron houses a secret chamber in the right tower where Odin is fought after he is struck down by Kainazzo. In Final Fantasy VI, Odin is petrified by a nameless sorcerer, and a hidden passage in the left wing of the castle leads to a chamber where he can be upgraded into Raiden. The same passage also contains the Blue Dragon - although the identity of Odin's opponent is not revealed, if it were the Blue Dragon that defeated him, this would place the story of his defeat in direct parallel to his death at the hands of Kainazzo. In addition the architecture of the two is similar.

  The world maps of the two games also share many similarities. Baron Castle is located where Figaro Castle would be, which fits the location of the Ancient Castle lying under Figaro's burrowing route. Doma Castle is in the north-east of the world like Fabul, and both are connected to the main continent by a narrow chain of land. Mt. Crescent lies where a chain of islands exist in Final Fantasy IV - if there was a flood, the mountain range would become a chain of small islands. Mt. Ordeals also stands in roughly the same location as the Cave to the Sealed Gate, and if the Warring Triad are somehow Lunarian in origin as mentioned above, it would make sense for Mt. Ordeals to be in the location of the Cave to the Sealed Gate, as it houses the spirit of KluYa and also is near the resting place of the Lunar Whale.

• Final Fantasy VII and Final Fantasy X/X-2: This is my favourite connection because I never realised it at the time, even having played them in chronological order. This connection was also explicitly told by one of the game's directors (Kietse I think). When you go to the Farplane, there is an Al Bhed boy called Shinra who talks about harnessing the energy of the farplane (much like Shinra using the lifestream). Although the connection is confirmed, how the connection plays out isn't. It's assumed that Shinra harnesses the energy and migrates to another world (being that Spira and Gaia are separate worlds), they are the Cetra from FF7.

  Note: Interviews with scenario writer Kazushige Nojima and producer Yoshinori Kitase in the Final Fantasy X Ultimania Ω and Final Fantasy X-2 Ultimania guidebooks, have revealed that Final Fantasy VII and Final Fantasy X-2 share a plot-related connection, in which the Shin-Ra corporation in Final Fantasy VII is founded by descendants of Shinra of the Gullwings in Final Fantasy X-2.

  Specific quote: Nojima (scenario writer for FF7, FFX & FFX-2)

  "Actually, it does. After quitting the Gullwings, Shinra received enormous financial support from Rin, and began trying to use Vegnagun to siphon Mako Energy from the Farplane. But, he is unable to complete the system for utilizing this energy in his generation, and in the future, when traveling to distant planets becomes possible, the Shin-Ra Company is founded on another world. That would happen about 1000 years after this story."

There are many more inter game connections, but many aren't confirmed. You can find a list here

LessPop_MoreFizz answer is correct. Each Final Fantasy is separate, even if they share some common points.

I would only add an exception: Dissidia and Dissidia Duo Decim.

Both Dissidia games link all universes in one big war between Light and Darkness. Each Final Fantasy game could be interpreted as one battle among many, where Dissidia is the epic conclusion of the war.