From Diablo Wiki
To twink is to hand down a high quality item to a lower level character, generally an item that this character would never have found for themselves at their experience level. The term also refers to a character that has been twinked as being a twink, and someone who is giving away items like this is twinking.
Twinking is usually done by experienced players in order to help their new characters level up more quickly.
Some players, often referred to as purists, do not twink at all, or at least not to certain characters. Players might want to keep their new character pure, or they might be playing with some severity of LotL rules, and are only using items they find themselves.
Twinking is sometimes considered a cheesy technique, but it becomes fairly ubiquitous amongst the low-level characters created by long time players of a game, and is a largely accepted fact of online play. Online buddies will often help twink, or rush a friend's new low level character.
In Diablo 2, it was useful to create and stock a twinking mule; one storage character with all sorts of low level gear. That made it easy to find the items you wanted to temporarily use on a new character, and it was quick to load them up with good starter gear, then reclaim it and exchange it for higher level goods as the new character leveled up.
Item in Diablo III require certain levels to equip, thus limiting twinking. Ways around this restrictions were added. Patch 2.1 introduced the Gem of Ease that, when leveled to 25, removed the level requirement of the weapon it was socketed in. Patch 2.3 introduced a recipe, through Kanai's Cube that removed the level requirement of any item, at the cost of consuming a level 25 Gem of ease. Both the Hellfire Ring and Hellfire Amulet roll without level requirement.