@Peter - You'd be looking for media queries. It allows you to set styles for elements based on the screen size. See more at the link below:
I don't understand much of what the link says:)
When I add the code that Caroline mentions, then I can easily exclude my image from mobile screens - but I don't know how to exclude it on tablet screens as well. Is it not possible to exclude for both tablet and mobile screens without changing or adding to the code provided by Caroline?