You don't download it from the Wine website. Wine is a Linux program that happens to work on Mac. So you get it the same way Linux users get their programs- through a package manager. You have two options with this:
Macports and
Fink. Macports is run by Apple and has much more up-to-date packages. But, it uses the old ports system- it downloads and compiles everything from source so it can take hours to actually install the program. Wine releases a new version every 2 weeks, so it's a bit of a pain to do this. Also, Macports doesn't have a stable GUI so you're almost certainly going to have to do this from the command line. Fink is a Mac version of the Debian Package Manager, which uses binaries so it's much faster but it's also not as up-to-date.
As far as which version to choose (stable vs. development), go with development. There aren't usually any regressions, and the "stable" is kind of an arbitrary point. For instance, version "1.0" came out after 15 years of development, and the marker for what would be version 1 is that Photoshop CS 2 should work near-perfectly out of the box. Right now, they're actually working on releasing another so-called-stable version, 1.2, so they're busy fixing the few regressions they did have.
Oh, and when you use the package manager, you'll want to get the wine-devel package. In Fink, just find the package in Fink Commander. In Macports, open the terminal and type "sudo port install wine-devel".