The 4th dimension in our case is where the 3D structures including this very Universe combine and exist within changing time frames. 4D structures can’t exist within 3D ones but 3D structures can exist in a 4D just like your drawings exist within that flat paper as lines and points but couldn’t exist in our 3D world by itself. Extra dimensions work the same, like a Matryoshka doll that loses and or gains properties the further you go.
Image: 3D projection of a tesseract undergoing a simple rotation in four dimensional space.
In mathematical physics, Minkowski space or Minkowski spacetime (named after the mathematician Hermann Minkowski) is the mathematical space setting in which Einstein’s theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime. [**]
In physics, spacetime (also space–time, space time or space–time continuum) is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as existing in three dimensions and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions. From a Euclidean space perspective, the universe has three dimensions of space and one of time. By combining space and time into a single manifold, physicists have significantly simplified a large number of physical theories, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels. [**]
But my favorite explanation of extra dimensions in general is Carl Sagan’s version. His version was based on Flatland: A Romance of Many Dimensions which is an 1884 satirical short story by Edwin Abbott Abbott:
The story is about a two-dimensional world referred to as Flatland which is occupied by geometric figures. Women are simple line-segments, while men are polygons with various numbers of sides. The narrator is a humble square, a member of the social caste of gentlemen and professionals in a society of geometric figures, who guides us through some of the implications of life in two dimensions. The Square has a dream about a visit to a one-dimensional world (Lineland) which is inhabited by “lustrous points”.
He attempts to convince the realm’s ignorant monarch of a second dimension but finds that it is essentially impossible to make him see outside of his eternally straight line.
He is then visited by a three-dimensional sphere, which he cannot comprehend until he sees Spaceland for himself. This Sphere (who remains nameless, like all characters in the novella) visits Flatland at the turn of each millennium to introduce a new apostle to the idea of a third dimension in the hopes of eventually educating the population of Flatland of the existence of Spaceland. From the safety of Spaceland, they are able to observe the leaders of Flatland secretly acknowledging the existence of the sphere and prescribing the silencing of anyone found preaching the truth of Spaceland and the third dimension. After this proclamation is made, many witnesses are massacred or imprisoned (according to caste).
After the Square’s mind is opened to new dimensions, he tries to convince the Sphere of the theoretical possibility of the existence of a fourth (and fifth, and sixth …) spatial dimension.
The depiction above is a 4 dimensional figure as represented by 3 dimensional cubes within cubes to visualize how 4th dimensions may work.
Related: Carl Sagan explains extra dimensions