Depth of field is an incredibly powerful tool, but the mathematics and mechanics of calculating and pre-visualizing it can be overwhelming.
A Polish software engineer (and amateur photographer) named Michael Bemowski recently put together one of the most helpful depth of field tools out there, and the best part is that it's completely free. The tool, which you can find here, allows you to manipulate every camera and lens setting that affects depth of field, from sensor size to focal length, from aperture to the distance between the subject and the camera. Plus it gives you a handy visual approximation of what each specific set of parameters would look like in a real world setting.
[Update: In the latest version of the app, the models themselves are no longer treated as flat objects. This means that you can see how focal length and depth of field can elongate or compress the human face. You can see a visual approximation of this here. There are a host of other new features in the update, most notably an Android app. You can read about the rest of the changes here.]
At the bottom of the app is another helpful tool, which gives you visual approximations of the distance between the camera, subject, and background, as well as the exact depth of field measurements for the settings defined above.
In order to make this tool as accessible as possible, you can download a version of it that runs offline on any operating system, and there is a dedicated mobile version as well, so you can access it anywhere at any time.