What image formats to use for basic 2d animation? In order to practice animation (and as a side project with my son) I want to create some basic 2D children animations using Adobe After Effects. The animations will have backgrounds, characters, movement, and speech.

Since my artistic ability is equivalent to that of a peanut, my artist friend will be providing the background and character files while I will be responsible for the animation.

In what format should I instruct the artist to deliver these files? PSD, vector, other? Is there a specific format that works best in After Effects?
That depends on what kind of animations you're planning. I would ask the artist to give you the project files (.psd if he is working in Photoshop, .ai if he is working with Illustrator) so that you are not limited to one final image (of course in that case, you would need said programs on your computer. Consider that if the artist uses CC and you're using CS6 or older versions, project files might not be compatible). If he refuses to send you the project files (which, in my experience, many graphic designers won't do), I would probably ask for high-res .png files (since they support alpha channels) if the artist's work is pixel-based. If the work is vector-based, ask for some vector format like .svg.