I have a Mac Mini that will boot to the Apple logo and sometimes to the spinning circle but no farther.
It is like that it starts to read the Hard Drive but then stops.
The hard drive is good, I checked it on another machine.
The memory is also good, I checked it on another machine as well.
I have a Snow Leopard disk but don't know how to use it to diagnose the problem.
The first think I'd do is run the hardware diagnostic that comes on the install DVD(s) that came with the Mac (note: these will not be available on a "generic" OS X disc bought separately from the Mac). Depending on what version of OS X your computer shipped with, put in either Mac OS X Install Disc 1 (through 10.5.4) or Applications Install Disc 2 (10.5.5 or later), then boot holding down the D key; see KB #HT1509 for details.
If you have AppleCare, that should have come with TechTool Deluxe, which you can use to run some additional hardware tests.
Assuming you don't find any hardware problems, I'd next turn to checking out data integrity on the disk. Boot from the install DVD (hold C this time, so it'll boot the installer rather than diags), select a language, and then run Disk Utility from the Utilities menu. Select your boot volume in the sidebar, then the First Aid tab, then the Repair Disk button near the bottom right. This will run an integrity check on the volume structure (i.e. it won't check the contents of the files, just make sure the data structures that keep track of them are consistent) and repair any problems it finds. If it finds problems, I'd recommend running it repeatedly until it gives a clean bill of health.
Next, while you're still on the First Aid tab, run Repair Disk Permissions near the bottom left. This sill set the permissions on the OS files back to what the install packages specified form them. Note that it only fixes the OS itself, not user files or even third-party software. Ideally, this should be run from within the booted OS, because there are some slightly version-specific tweaks, and the OS version on the DVD is probably different from what's on the hard drive; but it should get things 99.9% correct, and if that makes the system bootable you can then re-run it properly from the HD to fix anything the DVD version didn't get right.
If it's still not booting properly after that, try some alternate boot modes to get a better idea what's going wrong. I'd try verbose mode first: hold Command-V while booting, and (if it gets far enough) it'll go to a text interface where it prints information about what's happening as it boots. This won't fix anything, but may give an idea what's going wrong.
Next, try running the computer in Safe Mode by booting with the Shift key down (see KB #HT1564 and #HT1455) -- this disables noncritical system extensions, third-party daemons, and a bunch of other possibly-problematic system components. You'll be missing some functionality (IIRC sound and wireless networking both get disabled), but if you can successfully boot in this mode, you'll know the problem is coming from one of the things that get disabled in Safe Mode; play with disabling & reenabling them to isolate the problem further.
Finally, if that doesn't solve it, try booting in single-user mode by holding Command-S while booting. This interrupts the boot process early on, and drops you into a very minimal full-screen command-line interface. If you know a lot about how OS X works, you can dig around in this mode to find/fix problems, but at this point I'd just consider it a test: can the OS boot far enough to get to single-user mode? BTW, to get out of single-user mode, the standard thing is to use the "exit" command to continue the boot process where it left off; since that's failing for you, I'd instead use "shutdown -h now" to shut the computer off.
