As you grind your way through destiny 2 and collect Weapons and Armour items you end up with exotics and legendarys that you like to use but are stuck at the light level you found them at. Over time they will become less powerful than your light level.
With the option to infuse your items you can bring them up to a higher rating and keep them in your arsenal.
Infusing is the method of combining a wanted low level item with a similar one of a higher level that will be sacrificed. The sacrificed item is dismantled and removed while the item that you want to keep will attain the light level of the sacrificed item. The sacrificed item can be Rare (Blue) or above.
The process will also require one of more legendary shards as a cost. See How to get Legendary Shards.
In your character screen select the item that you want to upgrade in this way and press the option to view the details of the item. (Y - Xbox | Triangle -PS). In the items details screen the first square icon option below the name is the Infuse option. Hovering over this will give you a list of items in your inventory that you can infuse it with. If you don't have any items then you will need to collect more engrams from events and rewards.
Items will need to have a higher light level that the item you are upgrading and the same type (Armour / Weapon)
Simply hover over the unwanted item and press to dismantle it. And now you have an updated gun or Armour peice that you can continue using and grind for more rewards.