Dead Drops are audio recordings left by a government agent on the inside of one of the gangs. There are 32 Dead Drops in the game. You have to collect three of them in the Neon District for Mysterious Signals. The rest are optional and serve only to fill in further details of the story and earn a few trophies.
You can search for Dead Drops using your radar pulse. Press 
and watch the mini-map. If you are close enough to a Dead Drop, a gray ring will flash in a circle around the mini-map; or, if you are very close, a light gray ring will flash around Cole's icon (the white triangle). As the gray ring disappears, the final segment will flash and it indicates the direction of the Dead Drop. All Dead Drops are located inside small satellite dishes.