After backing up your system, insert your shiny OS X 10.5 DVD into your Mac and restart the computer, holding down C, so that the Mac boots off of the DVD-ROM.
After a few moments, you will be presented with a Leopard installation screen and presented with a number of options (some options may not appear on all systems, depending on model, previous OS version, etc.).
Upgrade to Mac OS X - this option will be fine for most users who are migrating from 10.4.10 and do not want to reinstall programs/personal data. If you are buying a new Mac that comes with the Leopard Upgrade pack-in - this will be the most hassle free choice.
Install Mac OS X - this is the option for anyone who either a) does not have another version of OS X on their system (new hard drive) or b) wants to do a new installation.
Under Install Mac OS X - there are several options:
Archive and Install
This option will archive the current contents of your system folder and install a new copy of Leopard. By default, your network settings (WPA or WEP passwords, VPN settings, etc.), user accounts and Home folders will be re-imported into Leopard. While third-party programs will be left intact, Apple still recommends that you reinstall them to ensure good working order.
Personally, Archive and Install isn't a bad idea - but if you already have up to date system back-ups anyway, and since you are going to have to reinstall most of your software, I say skip this step and just go for the second option, the option Zach asked about:
Erase and Install
This will erase the contents of your drive, format it into HFS+ and install Leopard clean. EVERYTHING ON YOUR HARD DRIVE WILL BE ERASED. If you choose this option, it is crucial that you back-up all of your data safely before the installation. If you select this option, you can click on an "Options" button that will give you additional choices, for instance, removing language files you may never use -- but if choices make you uncomfortable, just go for the standard installation.
After installing Leopard - whichever method you choose - it's time to re-import your data - using Migration Assistant, per Mat's guide.
Update: A lot of people in the comments are asking about boot camp updates -- unless you specifically use Disk Utility to delete or change your partitions, just installing on your set Tiger partition should leave your Boot Camp sector untouched. I'm not sure what changes were made in Boot Camp in Leopard from the beta - so you may end up having to reinstall or re-setup that partition. My Boot Camp partition was actually on another drive (using a cooked XP install with USB boot support enabled), so I don't know for sure.