You can easily fix it by clearing the Google Play Store and Google Services Framework from the configuration. To do this, go into settings and browse Apps> All Apps> Google Play Store> Clear data. Now, delete the data from Google Services Framework as well. This will assign a new ID from Google to your device, so your apps can act a little weird at first, but will not be a permanent problem.
Once you've cleaned the data, delete your Google account from your device and reboot. The link to your Google account is in Settings> Accounts> Add account> Google. Now you can download or update apps from the Google Play Store without problems.