Summary
They sleep with one eye open, one closed. Half the brain rests while the other controls breathing and watches for danger.
Dolphins sleep with half their brain while staying awake with the other half. They must breathe consciously - if fully asleep, they would drown.
When the left brain sleeps, the right eye stays open. After 2 hours it reverses. They sleep 8 hours total per day.
Newborn dolphins do not sleep for the first few weeks. Their mothers stay awake with them.
Source: Nature Neuroscience