Dolphins have brains quite similar in appearance to ours. Take the following image:
Notice the similarity between the dolphin's brain and ours, in size, and in texture. However, our brains work in very different ways.
The dolphin's brain structurally has two section, each with two lobes. Both of these lobes has a seperate blood supply, and in fact, can even be considered to be completely seperate brains. According to researcher Carol J. Howard, Dolphins never sleep with both parts of their brain simultaneously. In fact, when a Dolphin sleeps, one of the sides of it's brain completely shuts down, while the other stays alert in order to run the various body functions.
Each of the dolphin's eyes is connected to a different side of the brain. Because of that, a Dolphin can still see when it is sleeping, making it difficult for predators to sneak up on it.