What version of DNN was running on the site before the problem started?
Version 6. I don't know which version exactly.

Did you recently upgrade the DNN platform framework or a module. If so, to what version?
No

Did you create a backup of the site's files and database before attempting any upgrades?
Not relevant

Describe in detail how you added another site or DNN installation to the server?
First I had a URL-registration transfered to the hosting provider I'm now working with.
Then I tried to install the DNN-app that it provides again. (I think this was a mistake, I don't need a second DNN installation, I need another portal...)
Then I got the Install Wizard when I visited the website at the correct URL.
I had problems getting the Install Wizard to work, but I did manage to do so. I used the same database that I allready created for my first website. (Though I did create another database when trying to get the Install Wizard to proceed)
The installation didn't go succesfully, I got this error.
After this, my initial first website was down as well.

Do you have access via FTP to the site's files?
Yes

Do you have remote access to the site's MS SQL database?
No

Have you tried restarting the site's application pool on the server? Usually this can be done from your hosting provider's control panel, if not then by re-saving the site's web.config file on the server.
Yes, didn't help unfortunately.

Thank you very much for trying to help!

Ouch! That's what I was afraid you'd say regarding how you "added another site" to your DNN installation. Depending on how the hosting provider's DNN installer handled an installation over an existing site, you've basically done a failed upgrade from DNN 6.x to probably 7.1.x with the added problem that the site's web.config file was overwritten by that of the new install. Hence the reason that the install wizard appeared. My guess is that the database upgrade ran successfully before the install/upgrade failed.

At any rate, your best hope is to contact the hosting provider immediately to see if they have a backup of the installation's files AND database taken prior to your trying to add the second site. If so, it would be best to roll the site and database back to that from the backups. Most likely you will need help from the hosting provider to restore the database backup.

If the hosting provider does not have a backup of the site and database - and most likely they won't if it's been more than 2 or 3 days since the "upgrade" happened - we'll have to try to sort out exactly what has been broken and try to repair the site.

Here's some more questions and info needed if the hosting provider cannot restore from their backups:

1. Via FTP download the file dotnetnuke.dll from the site's /bin folder. On your local machine right click on the file, select "properties" from the context menu, and make note of  the file and assembly version #.

2. Via FTP examine the site's  /Providers/DataProviders/SqlDataProvider folder. Note the highest version number of the xx.xx.xx.SqlDataProvider files. Also look for xx.xx.xx.log.resources files. Any with xx.xx.xx greater than your DNN version # prior to the upgrade that are more than 1KB in size will contain errors logged during upgrade of the database.

3. Also via FTP look at the site's /Portals/_default/Logs folder for files that have names like InstallerLog#######.resources. Where ####### should be the date of the install/upgrade. These may contain other errors that were logged during the install/upgrade.

4. Do you happen to have a local backup of the site's web.config file that was made sometime (anytime!) before the problem occurred?

5. How critical is it to restore the site? Did it contain mostly text/html modules whose content can be recreated?