GNOME Shell got support for one click extensions installation starting with version 188.8.131.52, but the GNOME Shell extensions website wasn't ready until yesterday, when extensions.gnome.org has finally been launched as public alpha.
For now, you can only install extensions using Firefox, through a browser plugin that should be installed by default in GNOME 3.2. Also, the extensions installation (on/off) button only shows up if you're running GNOME Shell, so don't try to install an extension from Unity for instance, as it won't work.
There are already a few extensions to download: the official GNOME Shell extensions such as Alternative Status Menu or Dock, as well as some Frippery extensions like Bottom Panel or static workspaces.
According to the release announcement, extensions that use GSettings to store user settings cannot be installed as an user for now and a fix for this is targeted for GNOME 3.4.
Also, due to a bug in GNOME Shell 3.2.1, the uninstall button doesn't work for some extensions - you can disable the extensions, but to completely remove them you must manually delete the folder under ~/.local/share/gnome-shell/extensions/.
And finally: if the extensions require extra dependencies, you'll need to manually install them. Hopefully something will be done about this in the future.