A theory is a system of ideas. This can mean many things, depending on the context.
Math and science theories are testable explanations for events or occurrence.
Art theories are the conceptual structure behind the work. In day-to-day use, theory can mean an unproven idea or speculation, or a personal opinion: “In my theory…”
The word was first used in 1592, and meant (in general) a concept or scheme. By the 1630’s scientists used it in reference to a thought or explanation derived from systematic observation and testing. The verb form, theorize, also began to be used.
In Math and Science it is crucial for a theory to go through rigorous testing to become an acceptable explanation for a phenomenon. Of course, a theory can be debunked upon the discovery of new, conflicting evidence. Others are reinforced over centuries. For example, nobody doubts the truth behind Isaac Newton’s theory of gravity, which was developed in the seventeenth century.
Mathematical theory refer to bodies of knowledge in particular branches of mathematics. Though the complex explanations often fly over the heads of laypeople, these concepts are the building blocks of physics and other important sciences, and can help explain things like the movement of crowds to how the universe began.
Art theory can include art history or art criticism—anything which involves the analysis of work rather than creating the work itself.
For us lay people, theories are ideas, many of which are just rough ideas rather than well-researched systems of thought. It is quite common to shrug of an unfounded opinion with “oh, it’s just a theory.”