Ahhhh Ninja Gaiden or as here in Europe, Shadow Warrior. I still remember the first time i played it. I was twelwe and was visiting a friend of mine. He had a few NES games and my favourites of these were Faxanadu, Battletoads and of course Shadow Warrior! Sadly enough i never came longer than to the third boss. It was not until last year when i came across an import copy of Ninja Gaiden Trilogy that i was finally able to beat the game......
BARBARIAN
This guy is the games first boss and the first of the Malice Four. He slowly walks towards you, swinging his big cleaver thing. Since the guy is not exactly known for his great intellect you can actually just duck and slash at his legs. He wont even slash downwards! Just be shure not to get walked over by him.
BOMBERHEAD
Bomberhead is slightly stronger than Barbarian but not by much. He is the second of the Malice Four. He also has a wider range due to the chain and sickle he wields. He is pretty slow but the sickle can do heavy damage. You can destroy the sickle if you want but its probably a better idea to just kill him off as quickly as possible.
BASAQUER
This bloodthirsty psychopath is second in command only to Bloody Malph. He is alot stronger than the last to bosses in the meaning that he can jump! He is hard to hit and his constant bumping into you will drain your energy fast.... A tip: The Fire Wheel is a good bet against him.....
CERBEROS X2
These two dog thingies was once usual dogs but they were transformed by Jaquios evil power. He keeps them in the lower dungeons and they keep bouncing around the room shooting poison sludge at you. Just kill them off as fast as possible. 
BLOODY MALPH
Malph is the strongest of the Malice Four and he is also the warrior that killed Ryus father. He battles by throwing lightning from the sky. He can be quite hard if you are unprepared......
JAQUIO
Jaquio is the big bad guy of the game...... or so you think! He has proclaimed himself The Demon King and uses his magical powers to try to fend of Ryu. Can you beat him?
THE DEMON
This guy is the cause of all the trouble in the Gaiden universe..... I cant really give you any advice how to beat him. You have to find your own strategy......