Whenever I change my desktop background, it changes my laptop's background as well

To disable the synchronization of the certain settings to computers with your Microsoft account (on Windows 8):

1. Open the Charms bar, hit Settings, then choose Change PC settings.
2. On the left, click OneDrive.
3. Again on the left, click Sync settings.
4. In the main area, disable settings you don't want to sync.

The one that controls the desktop background is Appearance, the second one under the Personalization settings header. If you don't want any settings synced, disable the very top setting, Sync your settings on this PC.

On Windows 10:

1. Open the Settings app from the Start menu.
2. Click Accounts.
3. On the left, click Sync your settings.

The one that controls the desktop background is Theme here (source).

Changes to these settings take effect immediately.