There's a truckload of collectible items in Hitman: Absolution that you'll want to pick up throughout your playthrough of the game.
Unlike collectible items in other games, the ones in Hitman: Absolution make sense in the narrative of the game, as they come in the form of evidence items that you'll want to pick up, presumably to minimize your presence at the locations you go to.
Finding and discovering each of these evidence collectibles will net you a trophy/achievement called "Information is Power", in addition to a variety of in-game rewards like bonus points and instinct, which come in especially handy on the game's higher difficulty levels.
In addition to all that, evidence is also required to complete the challenges of each level in the game. You can keep track of your progress through the "Evidence Collector" challenge, which you can access through "Unlocks" in the Single Player menu.
When you collect a piece of evidence, it'll be saved to your character profile automatically so you won't have to complete the entire mission to keep the evidence. Simply quit out of the level once you get what you came for.
For an easier time finding evidence, be sure to use the Instinct ability, as it highlights nearby evidence.
Keep in mind that you'll have to play the game on normal difficulty or higher to complete challenges in the game. Easy mode won't allow you to keep track of your progress.
The video above by PowerPyx should help with that.
