The first thing to get out of the way is the movie starring Will Smith. Basically someone negotiated the rights to use Asimov's title, threw out all of his content, and made the movie they had already decided they wanted to make. (This is very similar to the movie Starship Troopers which shares a title and a couple of character names with Heinlein's novel, and nothing else. It would be better if they would just make their own damn movies, but at least some writers got paid, which I am always in favor of. I know a few writers. ).

This book is a collection of short stories originally written for the pulps in the 1940s. At the time, Asimov was always looking for story ideas, and had just read a book also called I, Robot by Eando Binder, and it helped spark an interest. He also has talked about how so many stories about robots took the view that they would turn into killers and would endanger us all. (Very similar to the breathless alarmism over Artificial Intelligence we see currently.) Asimov thought that was not the way things would work, so he wrote stories in which he introduced the Three Laws of Robotics:

First Law
A robot may not injure a human being or, through inaction, allow a human being to come to harm.
Second Law
A robot must obey the orders given it by human beings except where such orders would conflict with the First Law.
Third Law
A robot must protect its own existence as long as such protection does not conflict with the First or Second Laws.

These have entered the science fiction world as shared culture by now. And what they allowed Asimov to do was look for ways in which things might still go wrong, and then have his characters try to figure it all out. He later took his robots and the Three Laws into some Science Fiction/Mystery novels: The Caves of Steel, The Naked Sun, and The Robots of Dawn.

BTW, the real company U.S. Robotics took its name from the company in these stories. The real company is known primarily for making modems.