Many of our students in our English Skype classes have issues with these. Shadow is used to describe specific areas of darkness that are produced when objects or people are blocking any source of light. If you lift your hand up towards the light you create a shadow. It can be indoors or outdoors. A shadow can be cast by any light source such as a candle, a flashlight, an overhead light, a spotlight or the sun.
Shade, commonly used in a more general sense, is the darkness created by only the sun. Unlike shadow, it is an uncountable noun. The darkness underneath a tree or a parasol on a sunny day would be the shade. Shade is what one typically seeks on a hot sunny day in order to avoid too much sun exposure.