re: 1st time it failed the error was that the folder already existed.
That is a core WP error message saying the folder in which it is trying to install the Connections into already exists. You have to manually delete the folder by logging into your site with FTP and delete this folder:
re: I then downloaded a new copy of the file from your website and I just get the error that the install failed.
The plugin file downloaded from my site is the exact same one which WP will install as the link and the WP install link to the exact same file. I suspect since the folder already existed that the error was the same.
You can not install a plugin into an existing folder. This is a WP safety mechanism which is meant to prevent two plugins being installed into the same folder if they both happen to share the same folder name.
Seems someone accidently deleted the Connections plugin files and not its folder. As I mentioned it is not possible for Connections to self delete. OR…
Another possibility is that when WP was performing an update, the update failed part way thru. Part of core WP plugin update process is to delete the existing version. That is uncommon but does happen. This type of failure can happen to any plugin during an upgrade … OR … even an installation.
Hope that helps!