After choosing your options, click OK and then click Install, verify that you want to install, and enter an admin-level username and password. The installation will begin immediately--the computer doesn't need to restart first. If you're using a Bluetooth mouse or keyboard, you may have to re-pair it at some point, but otherwise, the next interaction you have with your Mac should be when it reboots with Snow Leopard successfully installed.
If you've set up your Mac to be able to boot into Windows via Boot Camp, you'll need to update the Windows drivers using the Snow Leopard DVD. Be sure to boot into Windows, insert the Snow Leopard DVD, and follow the instructions to update those drivers.
Faster and smaller
Apple claims that the Snow Leopard installation process is "up to 1.45x faster" than Leopard's. Indeed, installing Snow Leopard on a 2009 Mac mini took just under 29 minutes from the time I provided my username and password to the post-install restart--noticeably faster than installing Leopard on the same machine. (Restarting was faster, as well: the initial restart after installation, which is usually the slowest type of restart, took just over one minute.)
Apple also claims that installing Snow Leopard on a Mac running Leopard will free up "about 6GB" of hard-drive space, but you'll see this gain only if you installed a full version of Leopard, including all printer drivers. My testing on a 2009 MacBook Air that had only a subset of Leopard's printer drivers installed showed more modest gains: just under 2GB of "new" free space.
One of the other big changes the Snow Leopard installer brings is that it's much smarter about ferreting out possible software conflicts than previous OS X installers. For example, the installer disables most input managers, kernel extensions, mail plug-ins, and system hacks that are incompatible with Snow Leopard or are known to cause problems.