The reason it is so popular? It allows for infinite customisation and unlimited apps from third party sources that Apple wouldn't agree to. It can also be used to add stuff into other apps and integrate better as the developers are not limited to the API's that Apple provide in their SDK for iPhone.
The newest method of jailbreaking is a program called redsn0w, by the now world famous iPhone Dev-Team. Their tool is available for Mac OSX (10.4.11 and 10.5.6 upwards) and for Windows (XP and Vista). There is also called Pwnagetool, which allows for customisation of the firmware before you restore it onto the phone. For now, I will cover redsn0w as Pwnagetool is for MacOSX only and i don't have a Mac to run the program on.
Jailbreaking with Redsn0w on Windows
Now I only have a Windows box to test on but I imagine that the Mac way of doing it will be very similar if not identical.
I am assuming you are using an iPhone 3G with 3.0 firmware here. If you haven't got 3.0 on your iDevice yet, i recommend you do. If you are on an iPod Touch and don't want to pay for it, torrent sites are your friend. However, it is a fantastic update and I highly recommend paying for it (it is free on iPhone). If you wish to restore some firmware you have downloaded, Shift-Click the Restore button in iTunes and point it to your downloaded file.
BEFORE I START - DISCLAIMER
I am not responsible for anything that happens to your iPhone, iPhone 3G, iPod Touch 1st Gen or iPod Touch 2nd Gen. If you try this with an iPhone 3GS you are silly anyway, as it does NOT work with it. Be warned that using the jailbreak breaks your warranty with Apple - use it at your own risk.
Now
Download the most recent version of the program for either Mac OSX or Windows at the iPhone Dev-Team page (0.7.2 at time of writing).
Decompress the program (I use 7-Zip but the standard un-zip function in Windows does the same job).

Open the program. It will ask you to point it to the firmware. If you downloaded it from a different source to iTunes, you should know where it is. If you did download it from iTunes, you should follow the following instructions.

Open an Explorer menu (right click on the start menu and click on Explore)
Type into the Address bar %AppData%. Then navigate through Apple Computer > iTunes > iPhone/iPod Touch Firmware or something similar.
You could also copy and paste the following into your address bar:
D:\Documents and Settings\Administrator\Application Data\Apple Computer\iTunes
(if your user name isn't Administrator or your disk isn't D:, change these before you copy and paste it)

(I do not know what this would be on a Mac)
Plug in your device if you haven't already (Will work with everything apart from the new iPhone 3GS)
Then, follow the instructions. It will ask you whether you want Cydia (which is the main program for installing anything) and you do, believe me.
Make sure your phone is turned off. Then you will put the phone into DFU or Device Firmware Update mode. Follow the instructions and the exact timings as close as possible.
Power button for 3 secs
Power + Home buttons for 10 secs
Release Power but CONTINUE HOLDING HOME for up to 30 secs.
redsn0w will upload the jailbreak data and reboot your phone (takes about 2-3 mins). Do not try and hard-reset (holding down Home and Power for 3 seconds) or turn on/off your iPhone/iPod Touch in this time, because you could brick your device.
Congratulations, you are now on a jailbroken iPhone or iPod Touch.
Enhancing your experience on iPhone/iPod Touch with Cydia

Cydia is the program that will be used to install most programs on your iPhone/iPod Touch that do not come from the App Store (which still works, by the way).
To use Cydia, simply press it's icon on the Home Screen of your iPhone/iPod Touch. On it's first run, it will simply say "Reorganising" and it will "respring" your phone, which is like a restart but isn't, it just restarts the OS on the phone.
After it has done this, press it again. It will ask you whether you are a Developer, Hacker or User. I click Hacker, as I like having the command line tools. But most people will use User, as they just want the useful apps that are visual.

Recommended Apps:
1: Use the Featured Apps Section/SBSettings
Most of the time, there are lots of fantastic apps in the Featured Apps section. For starters, you will want SBSettings, which puts lots of useful toggles and little tweaks into a menu accessable at any time you can see the status bar. OpenSSH if you are a PC or Netatalk if you are a Mac are also vital packages. I will do a seperate tutorial for SSH'ing into the iPhone at a future date.

2: Winterboard
Winterboard is a hugely powerful application that can adjust nearly everything visual on your iPhone or iPod Touch. There are literally thousands of tweaks and adjustments and addons that are available on Cydia. These are put into action using Winterboard.

3: GRiS
Google Reader iPhone Sync is a fantastic, easy to use, quick app that is open source and runs on Python on the iPhone to bring Google Reader sync to the iPhone. Only useful if you use Google Reader.
4: Nes/Nes 3
Nes allows the user to play NES ROMS on the iPhone. While there are plenty of other emulators on the iPhone such as SNES, Games Boy and even MAME, the NES one is the only one I think works properly at a decent speed. However, I am only using an iPhone 3G and the iPod Touch 2G and the iPhone 3GS are much faster overall and so could probably run those fine. I cannot get NES to work on 3.0, but NES 3 is a paid-for alternative that has been optimised for the 3.0 software and SHOULD work. I haven't used so cannot confirm.
5: Terminal
Terminal is something that makes life easier for me, as someone who understands Unix command line, a lot easier as changing permissions and moving files can be done much more easily.

6: Cycorder
Only useful for the iPhone and iPhone 3G, this allows for recording video. Not fantastic quality and not completely full of features but better than the paid for alternative on Cydia, iPhone Recorder 3G and constantly developed by Saurik (again), the creator of Cydia (and Terminal and Winterboard and many of the other apps that are on Cydia).

Thanks to everybody I borrowed screenshots off (the iPhone Home Screen image and the winterboard screenshot). I do not claim them to be my own.
Enjoy your jailbroken iPhone or iPod Touch. Remember... I am not responisble for anything that goes wrong.