Update: the Snapshot tool is retired as of July 10, 2014. Any sites using it previously can continue to use it, but it will not be available for newer sites or sites that have not previously saved a Snapshot.
Start in the Customizer
To access the Customizer for your current theme, you can go to Appearance → Customize in your dashboard, or click the Customize button on the active theme listed in Appearance → Themes, or hover over your site or blog name in the main toolbar and select Customize from the drop-down list:
Create a Snapshot
Click Snapshots in the upper-right corner. Give your current settings a name and description so you will be able to tell one snapshot from another. (Dates are automatically included.) Click the Create button to save the snapshot.
The snapshot will be saved and appear at the bottom with the name and description you gave it.
Save Snapshots from Preview
If you’re deciding whether to purchase a premium theme or not, take advantage of the Snapshot tool. Start in Appearance → Themes and click Preview on a theme you would like to try.
Start in the Customizer, make changes to the various Customizer settings, and click Snapshots. Name and describe the snapshot, then click Create. The custom theme’s settings will be saved even though you have not purchased the theme yet. If you would like to exit without switching themes, do not click Activate or Purchase — click Cancel. To start using the theme now, click Purchase or Activate.
Applying a Saved Snapshot
At the bottom of the Snapshots view, one snapshot will show at a time. Use the arrow icons to bring the snapshot you wish to load into focus. You can work on it temporarily by clicking Preview. (If you make changes you wish to save, create another snapshot.) To load a saved snapshot and switch themes, click Apply.