Can we be conscious of our dreams? During the day time, we are conscious of our surroundings and the brain does its work.  

During sleep, we are not conscious. We have dreams. But the brain continues to function. After we awake though, there is only a memory of the dream experience. Is it possible to observe our dreams - that is, can we observe dreams as they occur and be conscious that we are dreaming?
Lucid dreaming is a practice in buddhism. La Berge (1980) describes in some detail how he learned to dream lucidly, and several authors managed to induce lucid dreams in their test subjects through electrostimulation, food supplements and other means. There are some popular books that claim to teach the skill, some of them written by La Berge.