1. Root - Root comes from the Linux terminology of Root access to your system, this is the top level controller of the operating system. IE in windows it's known as the administrator account. This lets you install/uninstall applications and change settings in your device that the carrier or phone manufacturer has locked down. A perfect example of this is people overclocking their Droids.
You have to use an SD race condition to access recovery console to gain root access to the incredible right now. Here is the full how-to. There are a few different ways to approach this but all do the same thing in the end.
public:adb_in_recovery [RootWiki]
2.NAND - The flash memory of the device. Specifically the flash memory of the device where you operating system, Android, and your "bios" or recovery partition, are located.
3.re-flash - Reflashing is the backing up of the default NAND area, and flashing it with a new recovery partition. This will enable you to get back into the root access area of your phone without much issue. It also allows you to run backups without needing to do any funky trips. IE NANDroid or Clockworks.
Think or the reflash as installing a new bios on your computer that gives you more options to setup your motherboard.
With all that being said, here is the laymans way of saying it:
You need administrator rights on your phone to access the parts of the memory that control how your phone operates and allows you to 'un-hide' the options that aren't available to you now. Once you get in there you will want to install a custom 'bios/loader' to enable you quick and easy access to those functions in an easy to use interface so you don't have to remember linux shell scripts to accomplish whatever it is you are trying to do.
Hope this helps a bit.
A few links of interest to "beef" up your terminologies.
NAND -
http://en.wikipedia.org/wiki/Flash_memory
This is what root mean! -
http://www.droidforums.net/forum/droid-hacks/632-what-root-means-stop-asking.html (A lot of good information here)
Incredible Root Process -
http://forum.xda-developers.com/showthread.php?t=699088 (100 pages and counting)
Linux Root definition -
http://www.linfo.org/root.html
Custom ROMs (Basically optimized Android O/S installs -
http://www.cyanogenmod.com/about (One of the most popular)
What is this 'adb' everyone keeps referring to! -
http://android-dls.com/wiki/index.php?title=ADB
What can 'adb' do? -
http://developer.android.com/guide/developing/tools/adb.html
Where can I go to read what other people are saying in the "dev" community? (Incredible specific) -
http://forum.xda-developers.com/forumdisplay.php?f=638