The world in which the story is set is mostly devoid of living animals. Consequently, owning an animal has become a symbol of wealth and status. Naturally, most individuals cannot afford such a luxury, and a whole industry of robotic animals has emerged to cater to the less wealthy segment who want to pretend to own an animal. The protagonist of the story even has a feud with his neighbour about the authenticity of their respective faux animals.
At some point, the protagonist comes in contact with a number of humanoid robots (androids). The theme of the book is very much about the blurry lines of what makes a living thing be alive (is the protagonist's own ultra-controlled life even called living?). So the title could be interpreted as "Do artificial humans have the same aspirations as us?", as in what's the difference between them and us.
The title is a reference to the common meditative practice of counting sheep to help one go to sleep. The implication is that androids, being electric, would possibly count electric sheep. The title implies a deeper question, that being whether machines are capable of dreaming, and if so, whether a machine that dreams can be considered human.
