The problem is that you don't have the ability to write (or save) to TeamSpeak's configuration files, which is located in Program Files (x86).
Windows Vista, 7 and 8 have a security system called UAC, which gives a security prompt before you can run certain files and save to certain locations (I'd include a picture, but I'm on a mobile device at the moment). People might suggest running the program as an Administrator to avoid file permissions, but I'd advise against it. Instead, during TeamSpeak's installation there's an option to specify where you want to save the configuration files.
You've already said that you've uninstalled TeamSpeak, so it's likely that you've already lost your IDs. For people who would like to retain their permissions across different servers, I'd advise doing the first step. If you don't care about saving your permissions, you can skip Step 1.
1. Open the Identities window (Settings >> Identities). Click on the ID you use - for many of you there is likely only one identity listed here - and press Export. Save the file somewhere; it doesn't matter where, only that you can find it later.
2. Uninstall TeamSpeak. If prompted, ensure that you check boxes or select YES to remove everything (again, currently on a mobile device so I'm uncertain of precisely what you'll be prompted with)
3.
NyanCaatt, I can tell from your Program Files directory that you're running a 64-bit system, so you can skip this step. For those who do not know the version of Windows they are running, determine the version you're running by clicking Start, right-clicking on Computer and selecting Properties. A window should appear. For Windows 7 you'll see "System type:" and whether you're running a 64-bit or 32-bit system.
4. Navigate to the TeamSpeak download page (
http://www.teamspeak.com/?page=downloads) and select Client 32-bit or Client 64-bit for your respective system.
See the next step for what to look for during the installation.
5. During the installation, you should be able to choose the configuration location (it occurs after you choose an installation location). Choose
Recommended: Save in user own files.
6. Complete the rest of the installation.
7. Open TeamSpeak. You'll be given a Welcome Wizard to help set things up. You can complete this or skip it, it's your choice.
8. If you saved your identities in Step 1, navigate back to the Identities window (Settings >> Identities) and Import the file you saved earlier. You can delete the Default ID and use the imported one now.
Using these steps, it
should work. For your case
NyanCaatt, the configuration files were being saved with the installation files. Every time you opened TeamSpeak it tried to save to the configuration file, but since it was in your Program Files (normally security-protected to some extent), Windows was denying TeamSpeak access.
Hopefully this fixes your problem! Let me know if it persists and I can look at the issue more in-depth when I get home! :]
Note: Make sure you restart your computer as well. It's a typical "Windows solution"