Durability is an attribute found on almost all items in LOTRO that determines how much an item can "wear" (usually determined by damage received, or by times used in case of tools) before it breaks and becomes unusable until it is repaired.
The amount of damage it takes before an item becomes broken is determined by the Durability value and the toughness (ie. Normal, Tough, Substantial). For example, an item with 100/100 Durability and a toughness of "Substantial" will take much longer to break than an item with the same Durability and a toughness of "Normal".
Some durability is also lost anytime the character is defeated after reaching level 10. Some special items reduce this durability loss.
Items can be repaired in most vendors.