Final Fantasy was released on the NES all the way back in 1987, when square was a small Japanese software company known for a fairly popular arcade game called Rad Racer. A US version wasn't released until 1990. At the time, nobody expected it to be anything special, as the RPG scene was dominated by the Dragon Quest series.
Thus Final Fantasy introduced an advanced character and gaming system to the console world for the first time. This combined with a rich plot resulted in a hit in Japan. This is the only FF game not to have chocobos or a Cid character.

Story:
Final Fantasy is set in a world where the powers of the four elements are bound and controlled by four crystal orbs. But these orbs are dying, and an evil darkness spreads itself across the globe The ancient prophecy of Lukahn is coming to pass, and only the coming of the four Light Warriors can save the world from evil forces.

Game system:
Unlike most FF games, you don't start with fixed characters, but select your party of four from six typical categories. This initial selection can greatly affect the overall difficulty of the game. A balance between fighters and magic users is usually the best option.
The in battle graphics are extremely simplistic, but all of the ingredients that render Final Fantasy battles what they are can be found here.

Guides:

Walkthrough
FAQ and Walkthrough
Instruction manual
-All 3 of above as zip archive (73K)

Square Artwork:

Cover Artwork
The Airship
A canoe
The 6 classes
A red mage and a knight
A fire cave
Titan
Garland
A Floating Fortress
The Light Warriors and a bridge

Can I play it? The original NES cartridge would be quite difficult to track down, but you can still play the game thanks to emulation.
Firstly, you need a copy of the cartridge: a ROM image (you can download these from hundreds of ROM sites)
To load the ROM on you computer, you just need a NES emulator like NESticle.