On death:
If your character is killed by an Other, they will come back as a Wight. If they're killed by a Wight, you get to decide if they do or not.
Their corpse will become the Wight, and your character will wake up whereever you like.
Symptoms of 'oh dear god this dead thing isn't going to stay dead is it' are that the skin remains ashen white, whereas the hands will be blackened. There might be frost in hair or on their features. The eyes will be almost glowing blue. And they'll probably try to attack you.
EDIT:
HERE is a concise list of known information about wights/others. Feel free to ask questions.
[whoo, I am so out of the loop today.]