A classic that many modern games owe thanks to
The biggest thing Gradius has going for it is the fact that you collect power-ups on the fly to upgrade your ship -- They can be weapon power ups that hover around your ship to add some extra umph to your firepower, or shield power-ups to save you from dieing a quick and painful death. Previous side-scrolling shooters really didn't have much in the way of these kind of power-ups, and if they did they were not executed nearly as well as Gradius pulled it off.
Not only did it have this but it had some excellent graphics at the time and great bosses awaiting you at the end of the missions, provided you actually survived the missions. Gradius isn't exactly a walk through the park -- it took time to master, and I can remember the controller being thrown on more than one occassion out of frustration for a particularly difficult stage.
Does Gradius hold up to the tests of time? Yes and no. You can see various incarnations of it as mini-games in other games (my personal favorite being in Legend of the Mystic Ninja on the SNES), and you can see its influence on cheap/freeware side-scrolling shooters all over. I'm not sure you'd care to go back to the original if you have access to Gradius III on the SNES, except for nostalgic reasons, or just to see what all the hype was about.