To finalize the installation process, you simply need to click the Upgrade button, which causes the installer to spend about 10 minutes "Checking Installation DVD." This step verifies the integrity of the installation disc and your hard drive before the installer actually does any installing. If you get an error message after this step, you may need to do some hard drive maintenance or order a replacement installer disc before proceeding.
Assuming all goes well, the installer will install Tiger by first "Preparing the Disk: Verifying Destination Volume" (which checks that you have enough free space, processor speed, and so on, as opposed to the earlier integrity check), and ends with "Optimizing System Performance." The whole process takes between an hour to an hour and a half, at the end of which you'll be prompted to restart your computer.
If you've left the computer alone to install and you don't notice the prompt, the installer program will automatically restart your computer after 30 seconds, so your computer will be all upgraded and ready to use by the time you get back from Starbucks. If you're watching as the computer restarts after installation, don't be worried when it takes a bit longer than usual or seems to stick at certain points. Remember that your operating system is re-creating all kinds of files that it will refer to on subsequent startups.