There are a total of 73 secret areas in Thief. Finding all of them will earn you the Finders Keepers Trophy / Achievement.
Whenever you enter a secret area you will hear a metallic sound in the background. In some cases you also need to interact with certain collectibles, objects or loot to trigger this sound. Do exactly what is shown in the video. If something is being picked up, you need to pick it up too. An area only counts when you hear the metallic sound! Be sure that music is enabled in the options and lower the Voice Volume & SFX Volume. This will make the metallic sound appear louder.

Unlike collectibles and loot, the secret areas are not listed in the statistics. There is no way to see how many you have already found. It's highly recommended to start a new game from scratch and to keep a check list. If you do indeed miss an area and don't remember which one it is, then you have to start all over again.

Also create a manual save game every time you discover a new area. The game does not save this progress automatically. So you either have to reach the next checkpoint or create a save game by yourself. If you die before saving the game, you have to visit the secret area again.

Here's a breakdown of how many hidden areas there are per chapter:
Prologue: 0
Chapter 1: 0
Chapter 2: 7
Chapter 3: 5
Chapter 4: 4
Chapter 5: 5
Chapter 6: 5
Chapter 7: 4
Chapter 8: 2
The City: 41