Are options a form of insurance? In my economics classes we have studied some really rudimentary concepts about insurance. I'm not really sure what qualifies as insurance to be honest. 

I was wondering if options are considered a form of insurance? Is that the right way to think about options? 

Perhaps a better way of phrasing the question is, "Is the primary purpose of options to provide insurance against changes in the stock's underlying value?"
No, the primary purpose of options is not to provide insurance against changes in the price of the underlying instrument: options don't have a primary purpose, they don't have an agenda, and they don't have a plan. They're just another tradeable instrument.

Some people buy them as means of insuring a position. Some people write them as a means of insuring a position.

And as Aurigae notes, in a crisis all correlations tend to one; so even if, in theory, your options insure the rest of your position, in reality, in a crisis, the counterparty risk can become very high, so the insurance is least available when it's most needed.