In the original Diablo, Nightmare and Hell difficulties were available that enhanced the toughness of the monsters, made many of them more resistant to magic effects, and provided greater rewards for treasure hunters. Diablo II is no different. Nightmare becomes available after Normal mode is beaten, and Hell difficulty is available once Nightmare is beaten. They will have many times the amount of health they had when you first encountered them in Normal mode. Monsters drop more gold.
Monsters give much more experience.
Unique and Super Unique monsters are given one extra attribute, i.e. Teleportation, Stone Skin, etc., apparently at random if a Super Unique. The maximum amount of gold for which items can be sold is capped at 25,000 gold. Mercenaries available in town are tougher, with higher levels and more damage. You will never lose a level because of this experience loss, so try and do anything really dangerous immediately after gaining a level. Nightmare and Hell are going to be where most high-level characters will be spending the majority of their time once they access them.