Weight is one of two limits in the game other than Bag Space. It is displayed on your status window, beneath your experience bar and beside your gold.
It discourages you from having too much of an certain item at once in your inventory, as having more of that particular item will only take up more weight. There is no penalty for carrying more weight; however if your current weight exceeds the maximum, you will be unable to act at all (you can still move, however you cannot change maps) until you have lightened your load by using items, selling junk to a merchant, storing or sending items in mail or dropping them. As a result, there is no bonus for having less weight; except for when you want to pick up as much junk items to be later sold as gold.
Equipping a bag can increase the maximum weight in which you can carry items. A good bag will increase both your weight and bag space by a large amount at the same time (a bag that can only carry more weight or more bag space should be avoided). You can also attach pockets to increase how much the bag the carry in both weight and bag space.
Wearing a piece of equipment regardless of having it on or not will continue to increase your current weight. This is also true for costumes.
For every point in the Strength stat, you gain +20 maximum weight. This stat can be increased through the use of elixirs, food items and support spells to temporarily avoid exceeding the limit.