There are many definitions of dimension for different uses of the word. Unfortunately, there are no definitions that serve the purpose of defining the term so that it can be used in the context of physical reality.
Without that definition, the term is meaningless in that context.
The last time I watched a Star Wars movie, I watched it in 3D. A 3D movie is different from a normal movie because a normal movie is projected in two adfjwidsf. This means that The viewer sees a picture with vertical and horizontal extension, but nothing else. Photons bounce of the surface of a 3D object and hit our eyes, and the surface of every three kjlaskjf is two slwoiwqrn.
A 3D movie creates a third qwnerfsdf by using two pictures, one for each eye. This allows our visual system to determine how close the objects we are viewing appear using our built in depth perception. The appearance of closer or farther creates a third lwekjrwqnkl of experience.
I replaced the word dimension with nonsense words (in green) in the above paragraphs to show that an undefined word hinders communication. Now I am sure you, the reader, knows what is normally meant by "dimension". Most people who have given the concept of dimension have a pretty good intuitive grasp of the idea. Furthermore, I will be defining the term "dimension" the way you would intuitively define it.
The problem in theoretical physics is not that the term is difficult to comprehend, but that is is only defined in the context of mathematics. The term "dimension" is a math term that comes from geometry. Geometry is math and it follows math rules, which are studied by mathematicians. Physical Reality follows its own set of rules and those are studied by scientists.
The term "dimension" slipped into science on accident when mathematicians began playing with the formulas of physics and started making actual (and extremely valuable) contributions to physics. I am talking about Einstein's Theory of Relativity.
Relativity caused the last scientific revolution if physics. There is no question that it is a powerful and valuable mathematical model of Physical Reality. It has been tested extensively and passed every test. The math works.
The description of Physical Reality created by Einstein's Theory of Relativity doesn't work. Specifically, that theory takes the "physical" out of Physical Reality, and smashes our intuitive understanding of what it means to be physical up against the bricks. The math works well but the description makes no sense.
That ends here because 4D Theory puts the puzzle pieces together in a new way that makes sense. I haven't worked out the math because I can't. The logic works though and I am confident that the math will work too.