Dream of the Endless

Dream is a member of the Endless, a group of living personifications of their name sake. Dream (as the name implies) controls dreams. However, in Gaiman's world, Dream also defines reality.

Morpheus

Morpheus is the first Dream of the Endless and is much older. Despite usually appearing to be a tall, “bone-white” skinned man with black eyes and unusually colored pupils, his appearance is known to change 'depending on who's watching'. For example, while Mister Miracle saw him as a man, Martian Manhunter saw him as Lord L'Zoril, a Martian god with the appearance of a flaming alien skull. However, when many gods were meeting with Morpheus, he appeared in the same form. To confuse the matter more, Bast, goddess of cats, said "I much prefer you in cat form, Dream old friend". So, it’s unknown how this appearance thing works. He likes to wear a helmet that is made out of the skull of a now-dead enemy of his, which is also his sigil to the other Endless. His speech bubbles tend to appear wavy and black with white lettering in normal caps (as opposed to the other character’s block caps).

Despite ultimately being heroic and responsible, he does have some negative traits. He tends to be cold, selfish and unwilling to accept change. The most extreme example of this was in “The Kindly Ones”, where he lets various denizens of the Dreaming (the heart of Morpheus’ domain) including Able, Fiddler Green and Merv Pumpkinhead, die in gruesome deaths instead of trying to save them (although he later let himself die to stop the Kindly Ones ). He is also usually blind to his flaws, but he can confront them if someone points them out. If an Endless loves a mortal, it will end badly. As a general rule, all of Morpheus’ romances end badly.

Prior the The Sandman series, it’s implied he is even more of a jerk. In “Preludes and Nocturnes”, he is imprisoned for 70 years and his kingdom falls apart. He escaped and after getting back his bag of sand, helmet and ruby, he started to rebuild his kingdom. In “Season of Mists”, he allows Loki to walk the world free. In “Brief Lives”, he kills his son (who was begging to be killed as he is now a bodiless head). In “The Kindly Ones”, Lyta Hall (who thinks Morpheus kidnapped his son, though Loki did it) released the Kindly Ones on the Dreaming and they go on a rampage (they’re able to do so because he killed his son). After several of his servants and friends die, he allows himself to be killed to stop them.

Daniel Hall

Prior to becoming the new Dream, Daniel was the infant son of Lyta Hall. His father was Hector Hall, whose ghost was manipulated by Brute and Glob (two nightmares) to become the lord of dreams. They did so in a forgotten corner of the Dreaming. Morpheus came and imprisoned the duo, “killed” Hector and sent Lyta back to reality. He later dubbed Lyta’s son Daniel. Loki kidnapped Daniel, but Lyta thought Morpheus did the deed and sent the Furies after him. Morpheus sent Matthew (A man now transformed into a raven) and the restored Corinthian (a nightmare loyal to Morpheus) to save him (which they did). When Morpheus was toast, Daniel (because he was a “child of the Dreaming”) became the new Dream.
The new Dream was a mix of Daniel and the entity of the Dreams. Unlike Morpheus’ speech, his was mostly normal, except for his word balloons being wavy. Notably, Daniel is much more nice and gentle than Morpheus. However, he is also inexperienced in some matters. So, he relies on the advice of the raven, Matthew, who worked closely with Morpheus.

References:
http://en.wikipedia.org/wiki/Dream_(DC_Comics)
http://en.wikipedia.org/wiki/Daniel_Hall
http://upload.wikimedia.org/wikipedia/en/b/bb/Dream.jpg
http://upload.wikimedia.org/wikipedia/en/a/af/Danielhall.jpg
The Sandman TPBs