ADB stands for "Android Debug Bridge". It comes with the android 2.0 sdk and can be run from the windows command prompt or a mac/linux terminal.

In order to run ADB from your machine, you will need to set up the following in your Droid "Settings".
Settings -> Application Settings -> Developement
Then check all of the boxes. Really, you only need the "USB Debugging" box checked, but I checked em all cause I'm cool like that.
Once you have ADB and have your Droid set for developement, you can plud your Droid into your machine with the USB cable supplied with the device.

On a Mac or Linux machine, you will have to "cd" to the "Tools" directory, under where ever you put the the "android-sdk-(mac or linux)" folder.
EX. On a mac you would do the following:
Code:
cd /Users/yourusername/android-sdk-mac/tools
(Its prettty much the same on a PC, but there are .bat files to run ADB for you in the "tools" folder)

Now that you can start adb here is how you can use it. I pulled the following from my terminal:
Code:
 Android Debug Bridge version 1.0.25

 -d                            - directs command to the only connected USB device
                                 returns an error if more than one USB device is present.
 -e                            - directs command to the only running emulator.
                                 returns an error if more than one emulator is running.
 -s <serial number>            - directs command to the USB device or emulator with
                                 the given serial number. Overrides ANDROID_SERIAL
                                 envivornment variable.
 -p <product name or path>     - simple product name like 'sooner', or
                                 a relative/absolute path to a product
                                 out directory like 'out/target/product/sooner'.
                                 If -p is not specified, the ANDROID_PRODUCT_OUT
                                 environment variable is used, which must
                                 be an absolute path.
 devices                       - list all connected devices
 connect <host>:<port>         - connect to a device via TCP/IP 
 disconnect <host>:<port>      - disconnect from a TCP/IP device

device commands:
  adb push <local> <remote>    - copy file/dir to device
  adb pull <remote> <local>    - copy file/dir from device
  adb sync [ <directory> ]     - copy host->device only if changed
                                 (see 'adb help all')
  adb shell                    - run remote shell interactively
  adb shell <command>          - run remote shell command
  adb emu <command>            - run emulator console command
  adb logcat [ <filter-spec> ] - View device log
  adb forward <local> <remote> - forward socket connections
                                 forward specs are one of: 
                                   tcp:<port>
                                   localabstract:<unix domain socket name>
                                   localreserved:<unix domain socket name>
                                   localfilesystem:<unix domain socket name>
                                   dev:<character device name>
                                   jdwp:<process pid> (remote only)
  adb jdwp                     - list PIDs of processes hosting a JDWP transport
  adb install [-l] [-r] <file> - push this package file to the device and install it
                                 ('-l' means forward-lock the app)
                                 ('-r' means reinstall the app, keeping its data)
  adb uninstall [-k] <package> - remove this app package from the device
                                 ('-k' means keep the data and cache directories)
  adb bugreport                - return all information from the device
                                 that should be included in a bug report.

  adb help                     - show this help message
  adb version                  - show version num

DATAOPTS:
 (no option)                   - don't touch the data partition
  -w                           - wipe the data partition
  -d                           - flash the data partition

scripting:
  adb wait-for-device          - block until device is online
  adb start-server             - ensure that there is a server running
  adb kill-server              - kill the server if it is running
  adb get-state                - prints: offline | bootloader | device
  adb get-serialno             - prints: <serial-number>
  adb status-window            - continuously print device status for a specified device
  adb remount                  - remounts the /system partition on the device read-write
  adb reboot [bootloader|recovery] - reboots the device, optionally into the bootloader or recovery program
  adb root                     - restarts the adbd daemon with root permissions
  adb usb                      - restarts the adbd daemon listening on USB  
  adb tcpip <port>             - restarts the adbd daemon listening on TCP on the specified port

networking:
  adb ppp <tty> [parameters]   - Run PPP over USB.
 Note: you should not automatically start a PPP connection.
 <tty> refers to the tty for PPP stream. Eg. dev:/dev/omap_csmi_tty1
 [parameters] - Eg. defaultroute debug dump local notty usepeerdns

adb sync notes: adb sync [ <directory> ]
  <localdir> can be interpreted in several ways:

  - If <directory> is not specified, both /system and /data partitions will be updated.

  - If it is "system" or "data", only the corresponding partition
    is updated.
From the above, you should be able to see that you can send adb commands from your machine to your Droid using the following:
Code:
adb -s [yourdeviceserialnumberhere] shell
The above will start an interactive shell from your machine, but running on your device. So if you "cd" to a directory, it will be on your device. Use "ls" or "ls -l" to see what is in the directory that your are currently in. NOTE: Runing the adb shell directly has gotten me nowhere as far as rooting the device, but its a good place to start learning.

Now for the fun part!! You can "pull" some data from your device using:
Code:
adb -s [yourdeviceserialnumberhere] pull /system /somedirectoryonyourmachine
This will try and write all the files and folders from the "/system" directory on your Droid, to wherever you decide you want it on you machine.

I did the "pull" on a mac and I was able to get about 140mb of the data from the following directories on my Droid:
/dev
/proc
/sys
/system
I did it both as an admin and a super user on the mac and I got about the same results. My Droid was in the regular boot mode. I have not tried booting into the restore or safe modes and tried a "pull" ... yet.

One last thing. I noticed the "adb root" command too, but I get an error back saying that you can not root on a production build of android. Perhaps there is a way to fool adb into thinking you have a developer device/build, maybe by altering the "build.properties" file. I haven't looked into that, nor do I really know if it would work. I'm really not an android developer, but if anyone out there is, maybe they could tell us what lets the adb shell know that a device is a production build, rather than a development build.

I better get going now. Have fun hacking your Droid!!