What is the difference between a theory in the common day language and a theory in the natural sciences today? Use examples please.
I believe that the difference lies in the degree of certainty. In common language, a theory can mean a proposal or an idea to explain a set of circumstances or observations. In this way, to theorize inherently means to conjecture. In other words, a person is trying to explain a phenomenon by coming up with sensible ideas that would lead to a sensible conclusion.
In science, we would more properly call this hypothesizing. A hypothesis could be thought of as a potential theory. But in the natural sciences, something could only be called a theory if it has been tested numerous times by different groups of people and found to be true each time. In the sciences, a theory is a time tested hypothesis that has proven itself to be extremely reliable such that it can be counted on in future situations.
So you see that the difference really between the two is the degree of certitude of the validity of the idea being proposed. Common language might use the the term in the sense of having a theory as to why the car won't start. The person doesn't really know why the car won't start, but they have a reasonable idea of why it won't start based on available facts and observations. They haven't really tested their theory by actually getting under the hood and taking the engine apart. Science on the other hand would use the term like with Einstein's theory of relativity. This was a hypothesis put forth by Albert Einstein that was eventually tested by independent groups at different times who ultimately all reached the same conclusion that it was true. So it was only called a theory over time as confidence in it grew with repeated testing.