Please bare with me on the screenshots, I could never draw anything more than rectangular circles and curvy lines in my whole life.

Update: Two more applications were added to the list, Tux Paint and Pinta.

Kolourpaint
Kolourpaint is a feature-rich painting program for KDE with a typical user-friendly interface, supporting many image formats (including PNG, JPG, TGA, PNM, GIF, BMP, XPM or TIFF). It allows zooming in/out, copy/paste, printing, cropping, rotating, applying various effects (balance, emboss, soften & sharpen). I think it is a perfect application for making simple paintings or sketches.

sudo apt-get install kolourpaint4

MyPaint
MyPaint offers many features and also an interface which is somewhat unusual among painting applications. There is the actual workspace and a separate window with predefined brushes to select from. It features usual functions, including undo/redo, zooming in/out, many brush types, several effects. Two noteworthy things to mention here are the existence of layers and also the brush editor, which allows you to create your own brushes.

sudo apt-get install mypaint

mtPaint
This application is full-fledged, and comes by default with a black background. It offers application-specific settings (like showing/hiding toolbars, splitting window), and also various effects, channels, or layers. Although it’s pretty good and has many options, I doubt it’s a good match for kids.

sudo apt-get install mtpaint

Gnome-paint
This is a very simple drawing application for the GNOME desktop with a typical interface and few options. It offers basic functions and two sidebars: one to choose the tool to use and the other one to choose the color. The feature to insert text is not yet implemented.

sudo apt-get install gnome-paint

Gpaint
Gpaint is yet another simple painting program from GNU, built with GTK, which offers standard functions and quite a few effects, like sharpen, emboss, oil paint or solarize. There is no undo function.

sudo apt-get install gpaint

Tkpaint
Tkpaint is written in Tcl/Tk and offers predefined shapes, various line types, text inserting support and several other tools. It can open GIF, PPM or BMP formats and will save the output image as Encapsulated PostScript or Tcl script.

sudo apt-get install tkpaint

XPaint
XPaint is rather a nice painting application, including a C script editor, toolbox, brush selector, font selector, color and pattern editor, and even a magnifier.

sudo apt-get install xpaint

Addition: Tux Paint
Tux Paint is especially designed for kids, and it provides bulleted buttons for the usual painting tools and brushes. The very nice thing about Tux Paint is that it provides various sounds when the paint tools are used or selected.

sudo apt-get install tuxpaint

Pinta
Pinta is yet another powerful application for GNOME, featuring zoom in/out, fullscreen mode, rotate, resize, brightness/contrast and hue/saturation settings, brushes and the typical painting tools. Except for those, Pinta also offers various effects and layers.

sudo apt-get install pinta