There are a total of 141 collectibles in The Last of Us. 85 Artifacts, 30 Firefly Pendants, 12 Training Manuals, 14 Comic Books. Finding all of them will earn you five trophies:
Look for the Light [Silver]
Endure and Survive [Bronze]
It was all just lying there [Silver]
I got this [Bronze]
Scavenger [Gold]

The game has 12 chapters. All of them (except for the prologue) do have collectibles. You can keep track of your total progress in the "Collectibles" section of the Bonus Menu, or by pressing the select-button while playing. After beating the game for the first time, you can also see in chapter select how many collectibles of each type you have found per chapter. You have to reach the next checkpoint after finding a collectible! Otherwise, the progress will be lost. You have to collect it again if you die.
All your collectible progress carries over to New Game+. You can also get missing items through chapter select.

Note: Some collectibles are placed in your inventory automatically and cannot be missed. These are not included in the video counter (because they do not have their own location in the game world).