The fourth dimension can be considered as being composed of time or space or any combination of the two. I call all the possible combinations the matrix. We happen to be trapped in the third dimension and we experience time and so if we think of time linearly we can interpret it as the fourth dimension. It just so happens that the world we inhabit is symmetry broken in a 3s+1t signature. But we can also interpret the fourth dimension as space, and think of it as just an extension of Euclidian space to the next dimension. In that case there would be no symmetry breaking. But I think it is also possible to think about four dimensions of time, which I call heterochrony. Dunne had this idea in a reaction to the work of Hinton in the twenties, and many authors tried to explore his idea, including Tolkein in Lord of the Rings. But why not two dimensions of time and two dimensions of space within a four dimensional configuration. I think all the permutations are possible.
What we should really be talking about is the unique features of the fourth dimension. That is what is going to tell you the most what it is. For instance the fourth dimension is such that all knots untie within it. It has six platonic solids while the third dimension has five and all higher dimensions have only three. All other dimensions have a fixed topology that is now known, but the fourth dimension according to Davidson has no fixed topology but rather an infinite number of fake topologies. The fourth dimension has quaternions as the basic rotational structure, and thus there is perfect rotation without any singularities as occur in the third dimension with equations that track motion. The fourth dimension is basically a mirror. If you rotate something through the forth dimension you get its enantiomorph, like when you turn a globe inside out and put it on the other hand. Basically all rotations in the fourth dimension are involutions in which the inside comes outside and vice versa. These are some of the interesting properties of the fourth dimension. When you take all these properties together and the constraints that bind them then you get the essence of the fourth dimension, which is its What that you are asking for.
But more interesting is the fact that the fourth dimension has intimate connection with the third dimension which also has some unique properties. Part of that is that the minimal solid of the fourth dimension the pentachora (pentahedron) has the same group A5 as the Icosahedron/Dodacahedron duals. Thus many of the properties that B. Fuller finds in the Platonic solids are in part due that there is an intimate relation between the third and fourth dimension. B. Fuller calls these properties Synergies and talks about them in his book Synergetics. Basically what I have been doing in my intellectual career is to extend B. Fuller’s analysis to the fourth dimension. And when you do that you find out amazing things.
Probably the most amazing thing, is the relation of the fourth dimension to nonduality. This is a very surprising development. Unfortunately it would take a long explanation to show why this is the case. That is because this is not an obvious fact but has to do with the structure of the hypercomplex algebras as a whole, and not with quaternions on their own per se. But you can get some flavor of an explanation at the site http://nondual.net.
So my answer would be that the true “What” of the forth dimension has to do with its nondual nature. It is our entry into the nondual, and it is just one dimension up from the one we are trapped within. If we interpret the fourth dimension as time we might not get this connection.