There's several types of demons:
- White-eyed demons are by far the most powerful. Lilith and Alastair are the only two known ones so far.
- Black-eyed demons are in general the weakest in terms of raw power and abilities. They're simply the most common.
- Crossroads demons generally have red eyes, and are far more self-serving than other types of demons.
- Of special note is Crowley, who was the leader of the Crossroads demons back when he first appeared. I had thought he was a black-eyed demon, but the SupernaturalWiki notes that he has never shown the color of his eyes. EDIT: See my answer here about Crowley.
And at least two that are unique with respect to their eyes:
- Samhain was a demon with a specific role, and his eye pattern is also unique - most of the eye turns white like the White-eyed demons, but the pupils stay black.
- Azazel seems to be the only yellow-eyed demon. It's been implied that he was king of Hell while Lucifer was trapped (although I don't think it was outright stated; he may have just had a high rank), and has been called the "tyrant who held the demons all together" by another black-eyed demon. More that I don't recall when it was revealed:
He shares the same resurrection powers as Crossroads demons, can possess Reapers, and was the one that not only discovered Lucifer was still alive, but also went about trying to free him on Lucifer's orders. Given that Lucifer created demons in the first place, I suspect Lucifer somehow managed to give Azazel a power boost that turned him into the only Yellow-eyed demon - but this is pure theory.
So, as an answer, there's two possibilities I can think of: Yellow-eyed demons are some sort of special "leader"-styled demon that generally never leave Hell (and are of a lower power level than White-eyed demons), or what I theorized in the spoilered section.
As of Season 12, we have an answer:
The Princes of Hell are the first generation of demons Lucifer created, meant to be the leaders of demon armies. All of them have yellow eyes.
By modern times, all of them except Azazel have lived on Earth and kept to themselves, staying out of Hell's business and avoiding the attention of hunters. Azazel alone was a "true believer" who still followed Lucifer, however one of them seems to be interested in Lucifer's offspring.