Most computers today use the address space to talk to the outside world (RAM, Disks, Networking, Displays, USB devices, Firewire devices, etc...).
The computer system designers generally have an chip on the motherboard that takes the address lines from the CPU chip and interfaces them with all the things the computer must talk to. On the Mac mini this is called the "North Bridge" and "South Bridge" chip sets. There is even a Wikipedia pages describing the North Bridge and South Bridge chip sets
<http://en.wikipedia.org/wiki/Northbridge_(computing)>
<http://en.wikipedia.org/wiki/Southbridge_(computing)>
NOTE: The Wikipedia articles are Windows PC specific, but the generic concepts are the same for a Mac mini.
The North and South Bridge chips used on the your's and my Mac minis can only handle 32 address lines, and 32-address lines has a maximum address range of 0 to 4,294,967,295 (0x0 to 0xFFFFFFFF or 4 billion in computer speak).
The Mac mini North Bridge Chip sends most of those addresses to the RAM chips you have installed (3,221,225,472 or 3 billion aka 3 gigabytes in computer speak).
The remaining 1 billion (1,073,741,824) addresses are used by the North and South Bridge chip sets to talk with I/O devices.
These remaining 1 billion addresses are mapped to control registers and device buffers in the various Graphic chips, USB controllers, Firewire controllers, and misc support chips used by the Mac mini (PRAM, SMC power controller, Time-of-Day clock (the wrist watch that keeps time when the computer is powered off, etc...)).
NOTE: The RAM chips NEVER see any signals from these last billion addresses. For example, you decide you need more seating in your home, so you buy 2 double length living room sofas (two 2GB DIMMs), however, your living room will only hold 1 and a half of your sofas (3GB), and the other half of a sofa is sticking out the door into your garage, and you cannot sit on that part because it is not properly supported underneath, and besides you use your garage for other things, such as cars, lawnmowers, snowblowers, garden tools, etc... (poor analogy, but hopefully you know what I mean). That last 1GB of RAM is just hanging around doing nothing, and cannot be used.
My point/question is that, if that "last 1GB of address space" were not there then the system would use memory out of the 3GB that is available for general use?
The CPU needs to steal address space to talk with its I/O devices and support chips. It does not need your RAM, just the addresses. If it only had 3 billion addresses, the computer system designers would use some of those addresses to talk to its I/O devices and you would not be able to address all of that 3GB of RAM.
Way back when, the early 8080 chips could only address 1 million addresses. The computer system designers gave 640 kilobytes to RAM and the remaining addresses to I/O devices. And when they first came out, we couldn't afford to put that much RAM into those systems :-)