I think the previous post does a nice job with the topic. I would like to submit that any potential storyline that examines the complexity of human emotion owes a debt to Shakespeare. In his plays and dramas, Shakespeare explored the issues of ambition, loyalty, honor, and the inevitable conflict that individuals face when trying to balance personal and public interests. From an intellectual standpoint, I would also suggest that Shakespeare is critical in providing a type of bridge between the ancients and the moderns. Consider that the definition of Ancient Tragedy, as defined by A.C. Bradley, revolves around the exploration and presence of equally desire able, but ultimately incompatible course of action. At the same time, the modern definition of tragedy became understood that every step of creation inevitably leads to destruction. Shakespeare was one of the few thinkers to compose Literature that encompassed both themes within both works and characters, making him a vital link to the examination of intellectual ideas both the past and present.
The major impact that Shakespeare has had on our society is in the fact that so many people are required to study his work in high school and college. Because we have all read or watched at least some Shakespeare, his works give us some amount of a common cultural background -- references that we all understand, etc.
Just think about all the Shakespeare references that are quite familar to us all. Everyone knows who Romeo and Juliet are -- if you call someone a Romeo everyone knows what that means. I just read a thing referring to a "star-crossed" quarterback -- another Romeo and Juliet reference. We all know about "to be or not to be," "shall I compare thee to a summer's day," and many other such quotes.
Shakespeare gives us something of a common set of cultural references and has, in that way, affected our society.