Are there any practical applications for higher dimensional geometry? My current understanding of higher dimensions is that we can take geometric properties from 2D or 3D, and simply extend them further theoretically. So in the same way that 3D objects can be projected onto a 2D space, 4D objects can be projected onto a 3D space.
But do higher dimensions "exist" in any sense outside of this type of theory? Is it just that we humans can't sense them, like we can't see infrared or hear ultrasound? And can we apply this theory to solve practical problems?
 A: Higher dimensions, and in particular, four dimensional spaces "exist" in the physical sense you describe. The study of such spaces is general relativity, which seeks to explain relationships between space, time, gravity, mass, and energy. General relativity explains the universe as a four dimensional space with three spatial dimensions and a time dimension. There are also other theories of physics which rely on the reality of ten or more dimensions, but I don't know too much about that.
But higher dimensional spaces are practical in other ways. One of my favorite examples is parameter space. Suppose you were designing a new cell phone. You might give it parameters, or numbers that describe it. For instance, you could let $w$ represent the width, $l$ the length, and $t$ the thickness. You could have $c$ represent the camera resolution, $m$ the memory, and $p$ the processor speed. We could go on, but that is sufficient for this example. With these numbers, we could represent the space of all possible cell phones by the ordered 6-element tuple $(w,l,t,c,m,p)$. Thus, these cell phones exist in a six dimensional space. While true that we cannot see this space, hold it in our hands, or draw it in a nice way, it is still a very real space. And it is far less abstract than most mathematical objects--ever element represents a cell phone one could build, hold, and use. Of course, cell phones aren't the only application. I know that biologists model certain processes that occur in cells in up to at least forty-something dimensional space. I once saw an analysis of how electricity was generated, moderated and distributed which employed a space with over one hundred parameters. My own research involves infinite dimensional spaces which nevertheless have real life application to the shapes of materials under stress. Thus, higher dimensional spaces can real and practical.
A: The question is asking about four dimensional space, specifically referencing tesseracts, and is not a general inquiry about higher dimensions. Although spacetime is clearly a way of understanding four dimensions and higher dimensions in general, this is not the question at hand.
It's absolutely necessary to point out this conflation, as Alex S. begins his answer saying that four dimensional space exists in a physical sense. I know he simply misunderstood the question, but this answer is so egregiously wrong that it can't stand.
Humanity has developed many ways of measuring physical quantities that we cannot or have difficulty perceiving, and these generally can be used to understand complex phenomena. Space, time, and temperature are at least five of the necessary dimensions for you to model the rate at which the heater on your stove warms up, for example. Alex mentions other examples where higher dimensions are useful.
On the contrary, four dimensional space is but a theoretical modelling abstraction, and there is no evidence that it exists in a physical sense. 
Even if we cannot perceive it like large portions of the electromagnetic spectrum, we can't measure it either. That doesn't mean that it isn't practically useful, but I'm not the right person for that. I've asked a friend to comment here on her work.
