Note: There are multiple ways of accomplishing this effect that will give different results. The method demonstrated here is just one quick, easy way.

Picking the Right Image

To get the best results, you’ll want an image that has clear, vibrant colors to begin with. We can always adjust them later, but good colors to start with will give us the best results. Also, since some of our work will involve painting by hand, the shapes you intend to color should be fairly simple with clear edges. If you wish, you can use this image which will be our example for the rest of this guide.

Once you’ve got your image, open it in Gimp. If you don’t have Gimp, it can be downloaded here for Windows, Linux, and Mac.

Layering

With your desired image open, move your mouse over to the Layers window and click the Duplicate Layer.

You should now see two identical layers. Make sure the top one is selected, go back to your image. To make this layer black & white, pick Colors -> Desaturate.

In the dialog box, make sure Preview is enabled and try out the different settings to find the one that looks best to you. When finished, your Layers window should now show both layers, but only the bottom one is colored.

Coloring Zones

This is the fun part. How this works is that we will be erasing the certain portions of the black & white layer to let the colored layer underneath show through, like a transparency on a projector.
If the image you’re working with is a JPG, you’ll have one additional step. In the Layers window, right-click your black & white layer and choose “Add Alpha Channel”. If the option is grayed out, then you already have an alpha channel and you’re good to go. Now, from the Toolbox window, choose the pink eraser tool (or hit Shift+E). Down below, you’ll have some options for your eraser. I’d suggest starting with a plain circle, size 7 or so. Fuzzy brushes can also be useful when doing this kind of coloring, but a solid circle will work for now.
We’ll start with the eyes, so it will be helpful to zoom in. You can do that from the View menu, or by holding Ctrl and scrolling your mouse wheel. When close enough, paint with the eraser over the portion you wish to color.

Alternate Techniques

You don’t necessarily need to hand paint every spot. If the section you want to color is a simple shape, you may be able to use the rectangle or ellipse selection tools, and hit Delete to remove the selection from the layer. The Fuzzy Select tool can be very helpful here as well if you adjust the Threshold value properly.
In particular, I recommend the Scissors Select Tool for this job. Click that from the toolbox (or hit I) and your mouse cursor should show a pair of scissors. This tool works by connecting dots that you specify, and try its best to properly wrap around the shape between those dots. Make the first dot by clicking anywhere you wish along the edge of the object you want to color.

When finished, click on the first dot again to complete the loop, and hit Enter to turn your series of dots into a normal selection. Hit the Delete key to remove the parts of the layer inside the selection.

Continue, using whichever tool seems appropriate for the task, for any objects you wish to color.