The name of the game

The name "Final Fantasy" could, in retrospect, be seen as a reference to the fact that no Final Fantasy game is truly a sequel to the last. The characters are introduced, and with your guidance they act out the tale, defeat the villain, and save the world - then we never hear from them again. Maybe their names will come up again, after all, there will always be another Cid and a new generation of Chocobos and Moogles, but once you get to the screen where you have to press reset*, these characters' story is over, forever. They have had their place in the history of their world, of the FF series, and in video games on the whole, and there is no more to be told.

That's not the real reason, though.

Once upon a time in the 1980s...
SquareSoft, the Japanese video game developer which is responsible for many, many great games, was running out of money. If their next title, a role-playing game unlike almost any seen before on a console, was a failure, then Square would probably go bankrupt. Square's employees saw the impending end of their employment and dissolution of the company, and thus they expected their newest game to be their last. Final Fantasy was a huge hit, however, and the name stuck.

Today Square even applies the title to games not originally designed for the FF series, because of the tremendous name-recognition it's acquired. For instance, IIRC, either the Final Fantasy Legend or Final Fantasy Adventure games for the Game Boy were actually from the Seiken Densetsu series. Also, Final Fantasy Mystic Quest for the SNES was actually from the Crap series.

* The screen where you have to press reset is a feature that many recent games sadly lack, even when they don't have second quests that would prevent it.


Difficulty

The original Final Fantasy is a difficult game. It does not have many of the luxuries and refinements of the later FF games. Many of the points of difficulty can be seen as features the game is lacking, or flaws in its design, but these things make the player's role in the game that much more complicated, and force the player to pay closer attention to many facets of the game that are largely automatic in more recent FF games.

Of course, this difficulty doesn't in any way make Final Fantasy I necessarily better than others in the series - much of the difficulty that I'm talking about is side-effects of decisions made by the game's designers, and not the results of a conscious effort to challenge the player. Many of these things WERE seen as flaws, and thus have not appeared in any FF game since.
Also, the game had an even worse, more ridiculous plot that recent FF games, and the characterizations and character development that make the recent games an absolute joy to play (until they get into metaphysical nonsense) are nonexistant.
FFI is still a heap of fun, though, and a good ol' 8-bit challenge.

Examples of Difficulty:

If you order two of your party members to attack the same enemy, and the first one kills it, the second character will not move on to the next enemy as they do in later games - they will make a strike against the fallen enemy, resulting in an "Ineffective" message instead of an amount of damage dealt. You can save a lot of time in battle if you keep track of the approximate amount of hit points common enemies have, and don't have your characters pile on an enemy that is near death, therefore preventing ineffective attacks.

You encounter random monsters as you travel in the ship.

In shops, you are not given a preview of how weapons, armor, etc. will affect your characters - you must buy items, equip them, and then see how it affects your stats. Careful management of equipment can save or ruin your game when, for instance, one of your mages runs out of MP, but will still deal an appreciable, though not exceptional, amount of damage - if you haven't neglected your mages' weaponry.

For most of the game, you'll never have enough money for all the magic spells you want - that's right, you have to BUY magic. You don't get to learn it while fighting battles, and it's not just given to you, ever.
There is no Phoenix Down, or any other item that revives dead characters and is available from the start of the game. If one of your characters dies, and you haven't had the opportunity to buy the LIFE spell yet, you have to get back to the nearest town and pay up to get the character revived.

The All-Mage Challenge

There is an incredible challenge to be had in FFI if you choose the correct party. By "correct" I mean "weak", of course. If you choose a party of all black or white mages, you will always be at a loss for good physical attacks. You'll always have the white mage's Masamune to look forward to, but by the time you get that, you'll be practically at the end of the game. A party of mixed black and white mages is actually a pretty good choice, but that kind of affirmative action just doesn't provide the kind of mindbreaking challenge you expect from the age of incredibly difficult NES games. It's easy to completely concentrate on magic with them, since you have a good balance between offense and defense, and don't have to worry about your white mages doing the former or your black mages doing the latter.

An all-black mage party is harder to deal with. You've got plenty of offense in whatever spells you decide to buy, but for healing you're stuck with potions. It's still a slow, hard fight, though, because your MP is very low at the beginning of the game, and potions cost money. It's possible to succeed with a party of those cute little shadowy guys, but it will involve a lot of tedious leveling.

An all-white mage party is even worse. You have plenty of healing, since you've got magic and potions at your disposal, but you have nearly no potential for attack. All you've got in the way of weapons to begin with is your stupid little mage stick, and IIRC there is ONE (1) white spell that does damage in the game. You will level, a LOT. You will die, a LOT. But if you beat Final Fantasy I with a party of four white mages, then you will have completed what may be the most challenging single trip through any video game. It's probably not worth it.

A friend of mine, David Amagai, has beaten Final Fantasy with four White Mages. He is a madman, by the way.