OK, listen up.
The code is one dimensional. Two dimensional if you're being pedantic and including time. The code itself is a linear sequence of data. You need one single piece of information in order to locate any part of the code when it's saved on your hard disk, or even when it's a program running in memory. That information is the memory/disk address. Your monitor receives a one-dimensional data stream (sort of). You think of the monitor as having, for example, 800x600 pixels. That's just an effect to make it easier to use. Really there are 480,000 pixels numbered 1 to 480,000. The rasteriser (a component of the monitor) breaks this long stream of info into lines so that it all fits on your monitor.
You think you're drawing to pixel (476, 396) when really the computer just sees that as the 316,476th pixel - that's 395 lines of 800 pixels each + the first 476 pixels of the 396th line.
Even your multidimensional array exists in one dimension. All memory is addressed from 00000000000000000000000000000000 to FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF in a 32 bit system. You can't use more pages of memory than you have addresses for. I know this is simplifying it a bit but it's still a one dimensional sequential address space. Notwithstanding how memory works - if you have a 5x5 array you still could have a one dimensional array with 25 elements. Do you want to know a secret? If you make a 5x5 2D array the computer makes a one dimensional array with 25 elements in it anyway! Only the computer remembers how you declared it. It knows to do a bit of maths with the index numbers. So element (3,2) becomes the 3rd element after the 2x5th element = the 13th element!
As for "dimensions" in general... someone said it before that a dimension is just a piece of information you need in order to locate something, like an axis on a graph. Imagine an ordnance survey map - it has 3 dimensions on it: Longtitude, latitude and height of the terrain in relation to "sea level". If you took a map like that and marked the population of every named location then you've added a fourth dimension.
In terms of physics then the convention is that a:
0 dimensional object is a self-contained point. It just is.
1 dimensional object is, essentially, a super string. The only way you can find a position on the string is by giving a distance from the start of the string.
2 dimensional object is a plane. You need an x and y coordinate to find all the possible locations on the plane.
3 dimensional object has width, height and depth as you well know.
The 4th dimension is time. A 4 dimensional object is one that exists in time. This means the other three dimensions can change over time.
After that it get tricky to imagine what the other dimensions mean. The other month in New Scientist some people reckoned that there are at least 17 different dimensions and that time, matter, gravity, the other fundamental forces (weak electric, magnetic and so on) happen when these dimensions overlap in some way. Really, this is cleary beyond humans to casually understand - it requires hundreds or even thousands of years of thinking and experimentation for us to work out what's really going on. Arguing that gravity isn't real or that the quantum uncertainty principle highlights how matter is a reflection of something else... it's just regugitating other people's arguments, and those people change their minds about things every month. Have you *read* a science journal over a long period of time? It's slow painful progress and people have a theory and persecute everyone that disagrees and then one day the other person is proved to be right and then a year later both are discredited... it's like following a politicians career.
This topic is some hideous chimera where you lot are tring to argue about computing, statistics/data handling and physics like it's all the same thing! Maybe the Grand Theory of Everything will prove that it is really all the same thing...