Wights are dead men or creatures raised up by the Others, seemingly when touched by the cold that accompanies them.  Anyone who falls against the Others must be burned, or else the dead will rise again as their thralls. Fear of their own dead becoming wights leads the wildlings to burn them.
The appearance of wights depends entirely on the condition of the corpse when it is raised. Some are lifelike, while others are badly rotten although the process of decay has been halted. All are easily identified by their eyes having turned bright blue  and their hands and feet black and swollen with pooled and congealed blood. 
Wights are supposedly attracted to warm blood and will attack with surprising strength.
It appears that wights retain at least some of their former memories, since the wight of Othor seemed to know where Lord Commander Jeor Mormont slept and who he was as it tried to kill him, and the wight of Thistle recognises Varamyr Sixskins even warged into the body of a wolf.
Being dead, Wights feel no pain and will continue to fight regardless of injury. Though they can be stopped by total dismemberment, their limbs will continue to move if detached from their bodies. When a wight is destroyed, the blue disappears from its eyes. 
Wights appear to be able to remain in a state of stasis, such as outside the hillside entrance to the cave of the three-eyed crow. The snows have buried most of the dead men, but they are still there, hidden, frozen, waiting. 
They are highly flammable and will be quickly consumed if set aflame. It is unknown at this time whether wights can cross the Wall on their own, although corpses brought through the Wall can apparently still reanimate as wights. Burning corpses appears to prevent them from being reanimated by the Others to become wights.
The Seven-Pointed Star
A list of individual characters that have been turned into wights over the course of the series:
A Game of Thrones
A Clash of Kings
Othor’s hand (still moving) was sent to King's Landing in a jar with Alliser Thorne to try to an impress on the king the severity of the threat from beyond the Wall so that the king may send more troops to the wall. Tyrion Lannister kept Ser Allister waiting so long that Othor's hand decomposed, and was not very convincing.
A Storm of Swords
During the Battle of the Fist of the First Men the Night's Watch was unable to defend against the wights that were sent against them. After a crushing defeat, the majority of the Night's Watch who fought on the Fist were slain by the insurmountable odds. Those who survived were led by Mormont on a grim march back to Craster's Keep.