What is the definition and implications of 4D space? By implications I mean if it existed how will it redefine what we know about physics and reality up to now. (Applications, possibilities, etc)
There are some current, active theories (such as string theory) that require as many as ten spatial dimensions.
Special relativity is a theory in 4D space. What does it change? There is a difference to the classical Newton mechanics which postulates only 3 dimensions. In newtonian mechanics we can rotate the coordination system to transform spatial axes one into another. Adding the fourth dimension (identified with time) allows us to "rotate" over new axes, that means to transform time dimension into spatial ones and vice versa. That means, one observer's time may be mixed with another observer's length. This "rotation" in the fourth dimension is exactly the Lorenz transformation. Lorentz length contraction and time dilation are the effects of looking at an object from different "angles" in the fourth dimension. Postulating more dimensions gives us just that: the ability to transform some physical quantity into another, specifically length and time. Suppose we postulate the fifth dimension and we identify the electric charge with the momentum in that dimension. So, there must exist a transformation ("rotation") transforming charge into length and time the Lagrangian is invariant under. These "rotation" transformations in higher dimensions usually have conserved quantities associated with them. Just as the angular momentum is associated with rotation transformation. Postulating the fourth dimension (time) gives us another conserved quantity associated with the resulting rotation group - the spin. That's why many authors write that the existence of spin is relativistic effect. If we postulate the fifth dimension identified with the electric charge, we also get a new spin-like quantity - the isospin. That's why I believe the Kaluza-Klein theories, but that's my personal preference. Postulating more dimensions implicitly assumes that translations in these dimensions are symmetries. This is not a problem with the time dimension, since it has been translation-symmetrical since the newtonian dynamics, but with fifth (electrical) dimension this means that there have to exist elementary particles with arbitrary high electric charge. This is called the Kaluza-Klein tower. The hypothetical elementary particle with 2e electric charge is called dilepton and it has been looked for.